THE  LIBRARY 

OF 

THE  UNIVERSITY 
OF  CALIFORNIA 

LOS  ANGELES 


The  RALPH  0. 


LIBRARY 


.MA 

LOS  ANGELES,  CALIF. 


TEXT  BOOKS  BOUGHT  &  SOLD 

COLLEGE    BOOK   COMPANY 
725  W.  6th  ST.  LOS  ANGELES,  CALIFORNIA 


-Or 


GENERAL   PRINCIPLES 


OF  THK 


METHOD   OF   LEAST  SQUARES, 


WITH  APPLICATIONS, 


DANA  P.  BARTLETT,  S.B., 

PROFESSOR  or  MATHEMATICS.  MASSACHUSETTS  INSTITUTE 
or  TECHNOLOGY. 


THIRD   EDITION. 


BOSTON 
THE  AUTHOR 
1915. 


COPYRIGHT,  1915. 
BY  DANA  P.  BARTLETT. 


TECHNOLOGY  BRANCH 

HARVARD  COOPERATIVE  SOCIETY 

76  MASSACHUSETTS  AVENUE,  CAMBRIDGE,  MASS. 

1933 


Geology 
Library 


PREFACE. 


The  preparation  of  this  volume  was  undertaken  with  the 
view  of  presenting  in  as  simple  and  concise  a  manner  as 
possible  the  fundamental  principles  of  the  Method  of  Least 
Squares.  While  it  is  believed  that  everything  essential  to 
the  solution  of  all  ordinary  problems  has  been  included,  no 
attempt  has  been  made  to  develop  at  length  those  special 
methods  and  forms  that  are  so  useful  and  almost  necessary  in 
case  large  numbers  of  observations  of  certain  kinds,  such, 
for  instance,  as  those  met  with  in  geodetic  and  astronomical 
measurements,  are  to  be  adjusted. 

Frequent  references  throughout  the  text,  and  more  particu- 
larly the  list  oi  works  given  on  page  v  of  the  Appendix,  will, 
however,  enable  the  student  to  extend  his  studies  in  what- 
ever special  direction  his  profession  may  require ;  it  being 
expected  that  this  book  will  in  such  cases  be  looked  upon 
merely  as  an  introductory  treatise.  All  of  the  works  men- 
tioned have  been  freely  consulted  in  the  preparation  of  these 
pages,  and  the  author  desires  in  particular  to  acknowledge 
his  indebtedness  for  many  of  the  examples. 

DANA.  P.  BARTLKTT. 


CONTENTS. 


CHAPTER   I. 
GENERAL   PRINCIPLES. 

PAGES 

§1.  Object  of  the  Method  of  Least  Squares.  —  3.  Errors.  — 
4.  Constant  Errors ;  Theoretical,  Instrumental,  Personal.  — 
6.  Mistakes.  —  6.  Accidental  Errors.  —  7.  Direct  Observa- 
tions ;  The  Arithmetical  Mean.  —  8.  Real  Errors.  —  9.  Resid- 
uals. —  11.  Weighted  Observations.  —  13.  The  General  Mean. 

—  16.  The  Curve  of  Error.  —  17.  Laws  of  Errors  of  Observa- 
tion. —  20.  Derivation  of  the  Equation  of  the  Curve  of  Error. 

—  22.  The  Method  of  Least  Squares 1-16 

CHAPTER   II. 

THE  ADJUSTMENT  OF  OBSERVATIONS. 

§23.  Indirect  Observations.  —  25,  27.  Rules  for  Forming  the 
Normal  Equations.  —  28.  Reduction  of  Equations  to  Weight 
Unity.  —  29.  Relation  Between  the  Weight  of  an  Observation 
and  its  Measure  of  Precision.  —  30.  Computation  of  Correc- 
tions. —  32.  Significant  Figures.  —  33.  Conditioned  Observa- 
tions. —  35.  Special  Cases.  —  36-43.  Empirical  Formulas  and 
Constants.  —  42.  Periodic  Phenomena.  —  43.  The  Logarith- 
mic Solution.  —  44.  Reduction  of  Equations  to  the  Linear 
Form 17-36 

CHAPTER   III. 

THE  PRECISION   OF   OBSERVATIONS. 

§48.  The  Constant  k.  —  49.  The  Value  of  k  in  Terras  of  h.  — 
61.  The  Mean  of  the  Errors,  or  Average  Deviation.  —  52.  The 
Mean  Error.  —  53.  The  Probable  Error.  —  55,  56.  The  Rela- 
tions between  p,  r,  a.d.,  h,  p,  and  p.  —  58.  Representation  of 
(*,  a.d.,  and  r  on  the  Curve  of  Error 37-46 


CONTEXTS. 

CHAPTER   IV. 
COMPUTATION  OF  THE  PRECISION  MEASURES. 

PAGES 

§59-61.  Direct  Observations  all  of  the  Same  Weight.  —  62.  Direct 
Observations,  the  Weights  Not  Being  All  Alike.  —  64-71.  Func- 
tions of  Independent  Observed  Quantities.  —  72.  The  Preci- 
sion of  Measurements.  —  74.  Functions  of  the  Same  Vari- 
ables. —  76-85.  Indirect  Observations.  —  76.  First  Method  of 
Computing  the  Weights.  —  77.  Rule  I.  —  78.  Second  Method 
of  Computing  the  Weights.  —  79.  Rule  II.  —  80.  Third  Method 
of  Computing  the  Weights.  —  81.  Rule  III.  —  82.  The  Mean 
Error  of  an  Observation.  —  85.  Observations  of  Unequal 
Weights.  —  86-89.  Conditioned  Observations 47-82 

CHAPTER  V. 

MISCELLANEOUS   THEOREMS. 

§90.  The  Distribution  of  Errors.  —  92.  The  Rejection  of  Obser- 
vations. —  93.  Criterion  for  the  Rejection  of  a  Single  Doubtful 
Observation.  —  95.  The  Huge  Error.  —  96.  Constant  Errors. 
—  98.  Combination  of  Determinations  having  Different  Con- 
stant Errors.  —  100.  The  Weighting  of  Observations.  — 
101-103.  Special  Laws  of  Error.  —  104.  Contradictory  Obser- 
vations   83-96 

CHAPTER  VI. 

GAUSS'S  METHOD  OF  SUBSTITUTION. 
§107.  Checks  on  the  Formation  of  the  Normal  Equations.  — 
108.  The  Reduced  Normal  Equations  and  the  Elimination 
Equations.  —  109.  Checks  ou  the  Solution  of  the  Normal 
Equations.  —  110.  Most  Convenient  Arrangement  of  the  Com- 
putations. —  111.  Application  of  the  Checks.  —  113.  Solution 
of  the  Elimination  Equations.  —  115.  The  Weights  of  the 

Unknown  Quantities 97-111 

GAUSS'S  METHOD  OF  CORRKLATIVES 111-116 

EXAMPLES 117-142 

APPENDIX. 

THE  THEORY  OF  PROBABILITY. 
§200.  Definition ;   Simple  Events.  —  202.  Compound  Events.  — 

204.  Dependent  Events i-v 

BIBLIOGRAPHY v-vi 

TABLES.     .  vii-xi 


THE  METHOD  OF  LEAST  SQUARES. 


CHAPTER  I. 

GENERAL   PRINCIPLES. 

1.  In  scientific  investigations  of  all  kinds  it  is  frequently 
necessary  to  determine  the  values  of  certain  quantities  by 
means  of  actual  measurements  either  with  or  without  the  aid 
of  instruments.  The  observations  may  be  made  directly  upon 
the  values  of  the  unknown  quantities  or  upon  certain  functions 
of  the  unknowns.  In  the  latter  case  the  values  of  the  required 
quantities  must  be  obtained  by  computation  from  the  observed 
values  of  the  functions.  In  order  to  obtain  more  accurate 
values  of  the  unknowns  than  would  be  given  by  a  single 
measurement,  or  set  of  measurements,  the  observations  are 
usually  repeated  either  in  the  same  way  and  under  the  same 
conditions  or  in  a  variety  of  different  ways  and  under  vary- 
ing conditions. 

Under  these  circumstances  it  will  invariably  be  found  that 
the  different  measurements  give  discordant  results,  the  amount 
of  the  discrepancies  varying  with  the  character  of  the  observa- 
tions ;  and  the  question  that  now  presents  itself  is  how  to 
determine  from  these  discordant  observations  the  true  values 
of  the  required  quantities.  From  the  nature  of  the  case, 
however,  we  can  not  expect  to  obtain  our  values  with  absolute 
accuracy;  all  that  we  can  hope  for  is  to  obtain  those  values 
which  are  rendered  most  probable  after  all  the  observations 
are  taken  into  account,  and,  further,  to  determine  the  degree 
of  confidence  that  can  be  placed  in  those  values. 


2  METHOD   OF  LEAST  SQUARES. 

2.  The  attainment  of   the  above   results  constitutes  the 
primary  object  of  the  Method  of  Least  Squares.     The  method 
is  also  employed  in  comparing  the  relative  worth  of  different 
measurements  of  the  same  quantity,  and  in  determining  the 
equation  of  a  curve  which  shall  suitably  represent  the  relation 
between  two  variables  in  cases  where  the  exact  law  connecting 
them  is  not  known. 

Also,  before  making  any  observations,  we  may  employ  the 
method  to  determine  how  precise  the  component  measurements 
of  a  series  must  be  in  order  to  yield  a  required  degree  of  pre- 
cision in  the  final  result;  or,  conversely,  to  determine  what 
the  precision  of  the  final  result  will  be,  knowing  the  precision 
attainable  in  the  component  measurements.  This  latter  appli- 
cation of  the  method  will  be  treated  at  length  in  the  course 
on  "  The  Precision  of  Measurements." 

3.  Errors.    The  cause  of  the  discrepancies  between  the 
results  of  our  different  observations  is  that  every  observation 
that  is  a  measure  is  subject  to  error.     These  errors  are  of  two 
kinds, — Constant  or  Systematic  Errors  and  Accidental  Errors. 

4.  Constant  Errors  are  errors  which  in  all  measures  of  the 
same  quantity,  made  with  the  same  care  and  under  the  same 
conditions,  have  the  same  magnitude,  or  whose  presence  and 
magnitude  are  due   to   some   fixed   cause.     These    constant 
errors  may  be   of   several  classes,  which  are  designated  as 
follows: — 

First.  Theoretical  Errors,  such  as  those  due  to  the  refrac- 
tion or  aberration  of  light,  the  effect  of  a  definite  change  in 
temperature  or  moisture  on  our  standards  of  measurement, 
etc.  As  soon  as  their  causes  are  known  the  magnitude  of 
these  errors  may  be  calculated  and  their  effect  eliminated 
from  the  observations. 

Second.  Instrumental  Errors,  such  as  errors  of  division  of 
graduated  scales,  defects  in  micrometer  screws,  eccentricity  of 
circles,  etc.  These  errors  will  be  discovered  by  an  examina- 
tion of  the  instruments  and  their  effects  eliminated  from  the 


GENERAL   PRINCIPLES.  3 

observations,   either  by  a    particular  method   of   using    the 
instruments  or  by  subsequent  computation. 

Third.  Personal  Errors.  These  are  due  to  personal  pecu- 
liarities of  an  observer,  who  always  answers  a  signal  too  soon 
or  too  late,  always  estimates  a  quantity  smaller  than  it  is,  etc. 
The  character  and  magnitude  of  these  errors  may  be  deter- 
mined by  a  study  of  the  observer,  his  "Personal  Equation" 
may  be  obtained,  and  his  observations  thus  corrected  for  this 
source  of  error. 

5.  Mistakes.    Although  of  a  somewhat  different  character, 
these  should  be  considered  in  connection  with  constant  errors. 
A  mistake  is  made  when  a  figure  3  is  read  for  a  figure  8,  or 
when  in  reading  a  graduated  circle  which  is  numbered  in  both 
directions  the  angle  is  read  43°  instead  of  the  complementary 
angle  47°,  etc.     These  mistakes  are  usually  of  such  a  charac- 
ter that  they  may  be  detected  by  an  inspection  of  the  observa- 
tions and  a  proper  correction  made. 

6.  Accidental  Errors  are  errors  due  to  irregular  causes, 
whose  effect  upon  the  observations  is  not  determined  by  any 
circumstances  peculiar  to  that  particular  set  of  measurements, 
and  which  cannot  therefore  be   computed  and  allowed  for 
beforehand.     Such  errors  are  those  due  to  sudden  changes  in 
refraction  owing  to  sudden  and  unobserved  changes  in  tem- 
perature; unequal  expansion  of  different  parts  of  an  instru- 
ment with  change  in  temperature;  shaking  of  an  instrument 
in  the  wind,  etc.     But  most  important  of  all  are  those  errors 
which  arise  from  imperfections  in  the  sight,  hearing,  and  other 
senses  of  the  observer,  which  render  it  impossible  for  him  to 
adjust  and  use  his  instruments  with  absolute  accuracy. 

After  a  full  investigation  of  the  constant  errors,  the  observer 
should  diminish  the  accidental  errors  as  much  as  possible,  both 
in  number  and  magnitude,  by  taking  every  precaution  and  care 
in  the  measurements  themselves.  The  problem  now  remains 
to  combine  the  observations  so  that  the  remaining  accidental 
errors  shall  have  the  least  probable  effect  upon  the  results, 


4  METHOD   OF  LEAST  SQUARES. 

and  it  is  to  bring  about  this  combination  of  observations  that 
we  employ  the  Method  of  Least  Squares. 

When  no  more  observations  are  made  than  are  sufficient  to 
determine  one  value  for  each  of  the  unknown  quantities,  we 
must  accept  these  values  as  the  most  probable  ones.  But  if 
additional  observations  are  made  leading  to  discordant  results, 
we  can  not  take  any  one  of  them  as  the  correct  value,  and  in 
fact,  as  already  stated,  we  shall  probably  not  be  able  to  obtain 
the  true  values  of  the  unknowns.  All  that  we  can  do  is  to  find 
values  of  the  unknowns  which  shall  remove  the  discrepancies 
between  the  different  observations  and  which  shall  be  those 
values  that  are  rendered  most  probable  by  the  existence  of 
the  observations  themselves. 

On  first  thoughts  it  may  seem  that  these  accidental  errors, 
being  due  to  so  many  different  and  unknown  causes,  will  be 
beyond  the  scope  of  mathematical  investigation.  Neverthe- 
less, the  theory  of  probability  requires  that  these  errors  shall 
follow  in  magnitude  and  frequency  a  law  that  is  capable  of 
exact  mathematical  expression,  and  experience  confirms  the 
correctness  of  this  law. 

For  more  extended  remarks  on  these  subjects  see  — 

Holman,  "  Discussion  of  the  Precision  of  Measurements,"  pp.  1-14. 
Merriman,  "  Text-Book  of  Least  Squares,"  pp.  1-6. 
Chauvenet,  "  Spherical  and  Practical  Astronomy,"  pp.  469-473. 
Wright,  "  Treatise  on  the  Adjustment  of  Observations,"  pp.  11-18. 


LAWS  OF  ERRORS  OF  OBSERVATION. 

7.  The  derivation  of  the  general  laws  of  the  occurrence  of 
errors  of  observation,  and  of  the  processes  for  determining 
the  most  probable  values  of  the  unknown  quantities,  will  be 
based  upon  the  following 

Axiom.  If  a  series  of  n  direct  observations,  M±,  M^  . . .  Mn, 
is  made  upon  the  value  of  a  quantity  M,  all  the  observations 
being  made  with  the  same  care  and  under  the  same  circum- 


GENERAL   PRINCIPLES.  5 

stances,  the  most  probable  value  M$  of  that  quantity  is  the 
arithmetical  mean  of  the  observations.     Or 


f  . 


8.  The  Real  Error  (a)  of  an  observation  is  the  difference 
between  the  observed  value  of  the  measured  quantity  and  the 
real  value. 

9.  The  l-tesidual  (w)  of  an  observation  is  the  difference 
between  the  observed  value  of   the  measured  quantity  and 
the  value  rendered  most  probable  by  the  existence  of  the 
observations. 

10.  Example.     Eight   observations  are    made   upon    the 
resistance  of   a  coil  of  wire,  the  true  resistance  being  512. 
Find  from  these  observations  the  most  probable  resistance, 
and  also  the  real  errors  and  residuals. 

Observations.  Real  Errors.  Residuals. 
M.                        x  v 

512.4  -f  .4  +  .30 

512.2  -J-.2  -J-.10 
511.9                    -  .1  -  .20        ^ 

512.3  -f  .3  -f  .20 

511.8  -  .2  -  .30 
512.3  -f  .3  -f  .20 

511.9  -  .1  -  .20 
512.0  .0  -  .10 

Mean  =  512.10  2v  =     ^00 

From  the  observations,  then,  we  should  say  that  the  most 
probable  resistance  of  the  coil  is  512.10.  It  will  also  be 
noticed  that  the  sum  of  the  residuals  is  zero.  That  this  is  a 
general  result  following  from  the  assumption  of  the  arith- 
metical mean  as  the  most  probable  value  may  be  proved  as 
follows :  If  the  observations  are  J/,,  J/j,  .  .  .  Mn,  the 


6  METHOD   OF  LEAST  SQUARES. 

arithmetical  mean  M^  and  the  residuals  v^  u2,  .  .  .  vn,  then 
we  have 

t?!  =  J/i  —  MO,       v2  =  My  —  MQ,      .  .  .  yn  =  Mn  —  M0. 

2,v='%lU~—  nM0 


=  2  M-  ?,M,     since  M0  = 


n 
.-.     Sv  =  0  (2) 

11.  Weighted  Observations.    The  weight  of  an  observa- 
tion expresses  its  relative  worth  compared  with  other  obser- 
vations.    Thus,  if  six  observations  are  made  upon  the  value 
of  a  quantity,  five  of  which  give  the  same  result,  while  the 
sixth   differs,  in   combining  these   two   different   results    to 
obtain  the  most  probable  value   of  the   unknown,  the  first 
value  ought  to  have  five  times  the  influence  upon  the  final 
result  that  the  second  has,  since  it  has  taken  five  times  as 
much  labor  and  time  to  obtain  it.     Hence  in  general  we  may 
say  — 

12.  The  Weight  (p)  of  an  observation  may  be  considered 
as  representing  the  number  of  times  the  observation  has  been 
repeated  and  the  same  result  obtained. 

The  weights  assigned  to  observations  may  be  due  to  a 
variety  of  causes,  as  difference  in  skill  of  observers,  difference 
in  the  instruments  used  or  the  circumstances  under  which 
the  observations  are  made,  etc.  But  whatever  the  cause,  the 
effect  on  the  final  values  of  weighting  an  observation  will  be 
the  same  as  indicated  in  the  preceding  paragraph. 

13.  Example.     Suppose  n  observations,  J/1}  M2,  .  .  .  Mn, 
of  weights  PI,  pz,  .  .  .  pn,    are  made  upon  the  value  of  a 
quantity  M.      To  find  the  most  probable  value    M0   of  the 
quantity. 

From  the  above  interpretation  of  the  meaning  of  weight, 
we  may  consider  that  the  whole  number  of  observations  is 
P\~\~Pz-\-  •  -  '  Pm  or  2/>,  and  that  the  result  Ml  has  been 


GENERAL  PRINCIPLES.  7 

obtained  in   />x   observations,    M2   in  p2   observations,  etc., 
Therefore,  by  (1) 

pl 

JWJ)   is  called  the  General  Mean. 

If  the  residuals  are  v^  v2,  .  .  .  vn,  we  have 


Q  (4) 

Which  shows  that  in  the  case  of  direct  observations  of  differ- 
ent weights  the  sum  of  the  weighted  residuals  is  zero. 

14.  If  the  observations  are   not  made  directly  upon  the 
values  of  the    required  quantities,  the  method  of  adjusting 
the  results  so  as  to   obtain  the   best  possible  values  of  the 
unknowns  will  depend  upon  the  laws  which  govern  the  dis- 
tribution of  the  errors  of  these  observations.     It  is  found  in 
practice  that  the  accidental  errors  of  observations  follow  cer- 
tain well  defined  laws,  and  what  these  are  may  best  be  seen 
by  taking  an  actual  example. 

15.  Example,     One  thousand  shots  are  fired  at  a  target 
which  is  divided  into  a  number  of  horizontal  sections  by  lines 
one  foot  apart,  the  centre   line  of  the  target  being  in    the 
middle  of  one  of  these  spaces.     The  shots  were  distributed 
as  follows  :  — 

In  Space.          Shott.  In  Space.  Shots.  In  Space.  Shott. 

!  -Hito+  i  19°  ~2i  to  -3£  79 

4  4-  i  "  --  i  212  -3^  «  -4£  16 

10  -  \  "  -1£  204  -4£  "  -5£  2 

89  -1     "  -2  193 


In  this  case  the  errors  are  evidently  the  distances  of  the 
shots  from  the  centre  of  the  target.     Further,  as  far  as  can 


8 


METHOD    OF  LEAST  SQUARES. 


be  judged  from  these  one  thousand  shots,  if  another  shot  is 
fired  the  probability  that  this  shot  will   fall   between   the 

lines 

.204 

•193 

«  .010 
«  .089 
«  .190 
«  .212 


—  £and  -: 

-ii  "  — ' 


«  .079 
«  .016 
«  .002 


-\-  5£  and  -f  4£  is  .001 
_^_4£    «    4-3i  "  -004 
+  3*    " 
+  2*    '< 

+  H    "    + 
+   *    "    ~ 

The  sum  of  the  above  probabilities  is  unity,  and,  therefore, 
as  far  as  the  preceding  shots  show  the  1001st  shot  will  cer- 
tainly hit  the  target. 

16.  Now  using  as  abscissas  the  distances  of  the  horizontal 
lines  from  the  centre  of  the  target,  and  as  ordinates  the  num- 
ber of  shots  falling  in  the  corresponding  spaces,  we  may 
construct  the  following  figure  :  — 


Figure  1. 


And  if  the  entire  area  of  this  figure  is  taken  as  unity  then 
the  area  of  each  rectangle  will  denote  the  probability  of  a 


GENEEAL   PRINCIPLES. 


9 


shot,  if  fired,  falling  within  the  corresponding  space  of  the 
target. 

The  graphical  representation  of  the  accidental  errors  of 
observation  will  always  give  a  figure  similar  to  the  above. 
Hence  denoting  errors  by  abscissas,  and  their  frequency  by 
ordinates,  the  law  of  error  of  any  series  of  observations  may 
be  represented  by  a  curve  whose  general  form  is  determined 
by  Figure  1.  This  curve  is  called  the  "  Curve  of  Error,"  and 
is  shown  in  Figure  2. 


/I 


// 

K' 


Figure  2. 


PDil 


In  order  that  this  curve  may  represent  exactly  the  distri- 
bution of  the  errors  in  any  given  series  of  observations  it 
ought  to  meet  the  axis  of  X  at  some  definite  distance  to  the 
right  and  left  of  the  origin  and  coincide  with  the  axis  from 
there  on,  for  in  all  actual  observations  there  is  a  limit  beyond 
which  no  errors  occur.  But  as  the  exact  point  of  meeting 
could  not  be  determined  for  any  given  case,  and  as  it  would 
not  l)e  possible  to  obtain  the  equation  of  such  a  curve,  we 
make  it  asymptotic  to  the  axis  of  X,  taking  care  that  the 
error  thus  introduced  shall  in  any  set  of  observations  be  so 
small  as  to  be  negligible. 


10 


METHOD    OF  LEAST  SQUABES. 


17.  An  inspection  of  Figures  1  and  2  will  now  exhibit 
some  of  the  general  lawc  of  errors  of  observation  and  the 
corresponding  properties  of  the  curve  of  error. 


Lavs  of  Error  derived  from  an 
inspection  of  Figure  1. 


Representation  of  these  laws  by  the 
Curve  of  Error. 


First.  Small  errors  are  more 
frequent  than  large  ones. 

Second.  Positive  and  negative 
errors  of  the  same  absolute  mag- 
nitude are  equally  likely  to  occur. 

Third.  The  probability  of  the 
occurrence  of  very  large  errors  is 
very  small. 

Fourth.  The  frequency  of  any 
error  depends  upon  the  magnitude 
of  that  error. 


The  maximum  point  of  the 
curve  is  on  the  axis  of  Y. 

The  curve  is  symmetrical  with 
respect  to  the  axis  of  Y. 

The  curve  is  asymptotic  to  the 
axis  of  X. 

The  equation  of  the  curve  will 
be  of  the  form 

if  ~  *^\       /  \       / 


18.  If,  now,  the  total  area  between  the  curve  and  the  axis 
of  X  be  denoted  by  unity  the  probability  that  the  error  of 
any  given  observation  will  fall  between  the  magnitudes  x  and 
x-\-dx  will  be  represented  by  the  area  included  between  the 
curve,  the  axis  of  X,  and  the  ordinates  of  the  curve  at  the 
errors  x  and  x  -\-  dx>  or  by 


y  dx  =  <f>(x)  dx 


(6) 


And  this  probability  will  be  known  as  soon  as  we  find  the 
form  of  the  function  <£(#). 

19.  The  above  expressions  in  (5)  and  (6)  are  the  ones 
that  we  should  use  if  we  regard  the  curve  of  error  as  repre- 
senting the  law  of  occurrence  of  errors  of  observation.  If, 
however,  we  look  upon  the  curve  as  expressing  the  law  to 
which  we  must  make  the  residuals  conform,  in  order  that  the 
values  of  the  unknown  quantities  obtained  from  them  may  be 


GENERAL  PRINCIPLES. 


the  most  probable  values,  we  should  replace  x  by  u  and  use 
the  expressions 

y=*(»)  (7) 


and  ydv  =  $(v)dv  (8) 

for  (5)  and  (6),  respectively. 

THE  EQUATION  OF  THE  CURVE  OF  ERROR. 

20.  Let  n  observations,  all  of  the  same  weight,  with 
results  MI,  MZ,  .  .  .  Mn,  be  made  upon  any  function  or 
functions  of  a  number  of  unknown  quantities  z^  z2,  .  .  .  zq  ; 
and  let  the  residuals  of  M^  M^  .  .  .  Mn  be  viy  v2,  .  .  .  yn, 
and  the  probability  of  the  occurrence  of  these  residuals  be 
<j>(Vi)  dv,  <£(v2)  do,  .  .  .  <f>(vn)  dv,  respectively.  Then  the 
probability  of  the  simultaneous  occurrence  of  all  these 
residuals  will  be 

.  .  .  <f>(vn)  (civ)*  (9) 


Each  different  method  that  might  be  adopted  for  computing 
the  values  of  the  unknowns  z1?  z2,  •  .  .  zq  would  lead  to  a  dif- 
ferent set  of  residuals  v^  w2>  •  •  •  yn  >  but  obviously  that  set 
of  values  of  2,,  z2,  .  .  .  zq  should  be  considered  the  best  which 
corresponds  to  the  particular  set  of  residuals  vt,  v2>  •  •  •  u«» 
the  probability  of  whose  occurrence  is  greater  than  that  of 
any  other  set. 

Therefore  the  most  probable  values  of  zt,  z2,  .  .  .  zq  are 
those  that  make  P  in  (9),  or  log/*  in  (10),  a  maximum. 

The  values  of  zu  22,  .  .  .  zq  corresponding  with  this  latter 
condition  are  those  that  satisfy  equations  (11).  It  maybe 
noticed  that  these  equations  also  express  the  preliminary 
conditions  leading  to  a  minimum  value  of  log  /J,  but  the 


12  METHOD   OF  LEAST  SQUARES. 

nature  of  the  problem  is  such  that  a  maximum  value  of  P 
evidently  exists  while  a  minimum  does  not,  and  it  is  there- 
fore unnecessary  to  investigate  further  the  mathematical 
conditions  for  a  maximum.  Hence  we  have 


,  __   «»n  _  0 


1      d  <£(Vi) 

f  ti    \  3<r  I      *    *    * 


(11) 

I        *     *    *   JL/M       \  T^f 


and  for  convenience  we  may  put 

S?=v 

substituting  in  (11)  we  have 


(14) 


These  equations  contain  all  the  unknowns  zt,  z2>  •  •  •  2?»  an(i 
there  are  as  many  equations  as  unknowns,  hence  as  soon  as 
we  find  the  form  of  the  function  u)  we  can  solve  these 


GENERAL  PRINCIPLES.  13 

equations  for  the  most  probable  values  of  zl5  22,  .  .  .  zq.  Since 
we  have  considered  the  general  case,  and  the  above  results 
are  to  hold  true  whatever  the  number  of  unknown  quantities 
and  the  form  of  the  functions  observed,  we  may  deduce  the 
form  of  \f/(v)  by  solving  a  special  example. 

Example.  Let  n  observations  of  equal  weight  be  made 
upon  the  value  of  a  single  unknown  zl5  with  results  M^  J/,, 
.  .  .  Mn,  and  let  the  residuals  be  vx,  y2>  •  •  •  vn.  Then  the 
most  probable  value  of  zt  is  given  by 


differentiating  with  respect  to  zx, 

dv!  dv2  dvn 

1        ..—         _  _     _     ^^_    _    —    i.  __  ._,  /  o   | 

dZi  ~  dZi  ~  3zt 

substituting  (a)  in  (14),  changing  all  the  signs,  we  have 

*0>i)  +  *("•)+•  .  .^(»»)  =  0  (b) 

But  in  this  case,  as  was  shown  in  (2), 

•»1  +   U2  +  '    •    •»»=    °  (C) 

In  order  that  (b)  and  (c)  may  both  be  true  the  functional 
symbol  i/>  must  indicate  multiplication  by  a  constant.  That 
is,  in  general 

!/r(t;)  =  cw  (15) 

Substituting  this  in  (13)  and  (12), 

dv 


dv 

therefore  — ^ =  cy  -^~ 

<j>(v)        dz  dz 

Integrating,  log  <£(y)  =  ^cy2-J 


I 


14  METHOD   OF  LEAST  SQUARES. 

Since  y=<f>(v)  is  the  equation  of  the  curve  of  error,  (7), 
we  may  therefore  write  it 


But  on  examination  of  the  curve,  y  is  seen  to  be  a  decreas- 
ing function  of  v,  and  hence  the  exponent  of  e  is  essentially 
negative.  Accordingly  we  will  write  our  equation  in  the 
form 

y=ke-h^  (16) 

the  values  of  k  and  h  depending  upon  the  character  of  the 
observations,  but  in  all  observations  of  the  same  kind  and 
weight  having  the  same  values. 

This  equation  represents  the  law  in  accordance  with  which 
the  residuals  must  be  distributed  in  order  that  the  best  results 
may  be  obtained  from  our  observations.  But,  as  before 
mentioned,  if  we  wish  our  curve  to  represent  the  most  prob- 
able distribution  of  the  real  errors  of  observation  we  should 
write  the  equation  in  the  form 

y=ke-***  (17) 

Hereafter  we  shall  use  without  further  remark  either  form 
of  the  equation  according  to  the  aspect  in  which  we  are 
considering  our  curve. 

An  inspection  of  the  above  equation  will  show  that  it 
satisfies  all  the  conditions  noted  in  discussing  the  form  of  the 
curve  of  error  in  paragraph  17. 

21.  It  is  important  to  notice  that  in  all  discussions  in  the 
Method  of  Least  Squares  the  number  of  observations  is  sup- 
posed to  be  large  and  always  greater  than  the  number  of 
unknown  quantities.  As  will  be  illustrated  later  on,  para- 
graph 91,  whenever  this  is  the  case  there  is  a  remarkable 
agreement  between  the  results  obtained  in  practice  and  those 


GENERAL  PRINCIPLES.  15 

indicated  by  the  theory.  And  even  when  the  observations 
are  few  in  number  the  method  still  affords  the  best  means  at 
our  command  for  their  adjustment,  the  results  obtained 
merely  having  a  smaller  weight  than  they  would  have  had  if 
derived  from  a  greater  number  of  observations. 

THE  METHOD  OF  LEAST  SQUARES. 

22.  We  are  now  in  a  position  to  see  whence  comes  the 
name  "  Least  Squares." 

In  paragraph  20  it  was  pointed  out  that  whenever  we 
make  a  series  of  observations,  each  observation  of  the  set 
having  the  same  weight,  the  most  probable  system  of  values 
of  the  unknown  quantities  will  be  that  which  corresponds 
with  the  set  of  residuals  the  probability  of  whose  occurrence 
is  a  maximum.  That  is,  the  best  set  of  values  of  the 
unknowns  will  be  that  which  gives  a  maximum  value  to 


But  from  equation  (16)  this  reduces  to 

P  =  kne~h<iW  +  v*  +  •  •  •  f»2)  (dv)*  (18) 

Since  the  exponent  of  e  in  this  expression  is  negative, 
evidently  P  will  be  a  maximum  when 

v?-\-v*-\-  .  .  .  i?«2  =  Sv8  is  a  minimum.     (19) 

Hence  the  adjustment  of  observations  by  the  Method  of 
Least  Squares  is  based  upon  the  principle  that  the  most  prob- 
able system  of  values  of  the  unknowns  is  that  which  renders 
the  sum  of  the  squares  of  the  residuals  a  minimum.  Hence 
the  name. 

The  conditions  for  a  maximum  value  of  P  were  expressed 
in  equations  (14),  and  since  it  has  been  shown  that  the 


16   i  METHOD   OF  LEAST  SQUARES, 

function  \f/  means  multiplication  by  a  constant,  those  equa- 
tions reduce  to  the  following,  called 

NORMAL    EQUATIONS. 


Vi    .  V2    .  vn 

a  —  4-  V2  a  --  h  •  •  •  V»  a  —  =  0 
~     Vz~  * 


An  inspection  of  these  equations  will  show  that  they  also 
express  the  conditions  that  will  make  the  sum  of  the  squares 
of  the  residuals  a  minimum. 

In  the  adjustment  of  observations  the  above  are  the  funda- 
mental equations.  In  order  to  obtain  the  most  probable 
values  of  the  unknowns  in  any  set  of  observations,  all  that  is 
necessary  is  to  form  the  Normal  Equations  for  that  set  and 
solve  them  simultaneously.  The  examples  already  solved  for 
direct  observations  are  merely  special  cases  of  the  above 
general  solution. 


CHAPTER  II. 

THE   ADJUSTMENT   OF   OBSERVATIONS. 

INDIRECT   OBSERVATIONS. 

23.  In  the  determination  of  the  values  of  quantities  by 
means  of  observations  the  functions  of  the  unknowns  that  it 
is  necessary  to  observe  may  be  of  any  form,  but  if  they  are 
not  linear  the  normal  equations  derived  from  them  are  likely 
to  be  complicated  and  difficult,  if  not  impossible,  to  solve. 
Hence  if  the  observations  are  not  upon  linear  functions  of 
the    unknowns,    the   first   step  will   be   to   reduce   them   to 
equivalent  linear  expressions  by  transformations  depending 
on  the  character  of  the  observed  functions ;  see  paragraphs 
43  and  44.     It  will  be  necessary  to  consider,  therefore,  the 
method  of  adjusting  observations  on  linear  functions  alone, 
and  the  procedure  in  cases  of  this  kind  may  be  illustrated  by 
the  following  simple  example. 

24.  Example.     S^  £2,  $3,  are  three  solids  whose  masses 
are  required.     Not  having  standard  weights  enough  to  obtain 
all  these  masses  directly,  by  varying  the  distribution  of  the 
solids  in  the  pans  of  the  balance  the  following  observations 
are  made :  — 

S^  =  Sz  -|-  1.7  grams. 
£,==  2.4      « 

£s+A',=  A\-t-1.0      " 
»S'2  =  8,  -f  3-°      " 

If,  now,  the  most  probable  values  of  /S^  St,  £„  are  repre- 
sented by  2,,  22,  28,  and  the  corresponding  residuals  of  the 
observations  by  v^  u2,  .  .  .  «4,  the  values  of  these  residuals  in 
terms  of  2,,  22,  28  may  be  found  from  the  above  observations 


18  METHOD   OF  LEAST  SQUARES. 

by  transposing  all  terms  to  the  first  members  of  the  equa- 
tions, and  we  obtain  at  once  the  following,  called 


OBSERVATION    EQUATIONS. 
zl  _  22  —  1.7  =  vt 

l'-li=l*        (A) 

22  _  23  —  3.0  =  v^ 
Applying  equations  (20)  as  formulas  we  have  the 

NORMAL    EQUATIONS. 


22+23  -1.0) 
(22-23-3.0)  =  0      (B) 


-f-(22-23-3.0)(-l)  =  0 

Simplified,  these  become 

22t  —  222—    23  —  0.7=0  (a) 

—  221-J-3«2  —2.3  =  0  (b) 

—  0.4=0  (c) 


3X(a) 

62!  — 

622  — 

32,  -  2. 

1 

=  0 

(c) 

—     2t 

+ 

323  —  0. 

4 

=  0 

52i  — 

622 

2. 

5 

=  0 

2  X  (b) 

—  4^  + 

622 

—  4. 

(> 

=  0 

«1 

7 

1 

=  0 

z.  j 

7 

.1 

(d) 

substitute 

(d)  in  (b) 

K-.J 

=  5 

.5 

(e) 

" 

(d)  in  (c) 

3» 

=  2 

.5 

(*) 

An  inspection  ,of  the  work  in  this  example  will  show  that 
for  the  adjustment  of  observations  of  equal  weight  on  linear 
functions  of  the  unknowns  we  may  derive  the  following:  — 


THE  ADJUSTMENT  OF  OBSERVATIONS.  19 

25.  Rule.     For  each  observation  write  an  "Observation 
Equation  "  /    then  for   each   unknown  form    a   "  Normal 
Equation"  by  multiplying  the  first  member  of  each  observa- 
tion equation  by  the  coefficient  of  that  unknown  in  that 
equation,  adding  the  results  and  placing  the  sum  equal  to 
zero.     Solve  these  equations  simultaneously  for  the  values 
of  the  unknowns. 

In  solving  for  the  most  probable  values  of  the  unknowns 
the  second  members  of  the  observation  equations  are  very 
commonly  written  zero  instead  of  v^  v2,  .  .  .  vn.  For  this  is 
the  form  in  which  the  equations  naturally  appear,  and  if  the 
observations  were  exact  the  residuals  would  actually  all  be 
zero.  The  method  of  solution  is  the  same  in  either  case. 

26.  Observations  of  Unequal  Weight.    If  the  observations 
are   not   all  of  equal  weight   the  same   method  will  apply, 
except  that  in  the  formation  of  the  normal  equations  each 
•observation  equation  will  be  used  the  number  of  times  denoted 
by  its  weight.     Thus  in  the  last  example  if  the  observations 
have  the  weights  4,  9,  1,  4,  the  normal  equations  will  have 
the  same  form  as  in    (B),  page  18,  but  each  part  of  each 
equation  will  be  multiplied  by  the  weight  of  the  observation 
equation  from  which  it  is  derived.     This  will  give  the 

NORMAL    EQUATIONS. 
4(21-22-1.7)  +  (-21  +  22  +  23-1.0)(-l):=0 

4(2l  -  22  -  1.7)(-  1)  +  (-  ai4-2a  +  2,  _  1.0) 


9(2,  -  2.4)  +  (-2l  +  22  +  23-1.0) 

+  4(28-2,-3.0)(-l)  =  0 
or  reduced 

52l  —  522  -       23—    5.S=0  (a) 

—  52!-(-922—    323—    6.2  —  0  (b) 

.    2l  —  322+1428—  10.6=0  (c) 

the  solution  of  which  gives 

z,  =  7.07       22  =  5.42       z:}  —  2.42 


•20  METHOD    OF  LEAST  SQUARES. 

Further  it  will  at  once  be  seen  that  if  p^  p2,  .  .  .  pn,  are  the 
weights  of  the  corresponding  observations,  equations  (20) 
take  the  general  form: — 

WEIGHTED    NORMAL    EQUATIONS. 

5^2    ,  9vH      A 

'-.nVn--- 


8vn 


Hence  for  the  formation  of  the  normal  equations  in  weighted 
observations  on  linear  functions  of  the  unknowns,  we  have 
the  following: — 

27.  Rule.     For  each  observation  write  an  "  Observation 
Equation "  /     then  for   each   unknown  form   a   "Normal 
Equation"  by  multiplying  thejirst  member  of  each  observa- 
tion equation  by  the  coefficient  of  that  unknown  in  that 
equation  and  by  the  weight  of  that  equation,  adding  the 
results  and  placing  the  sum  equal  to   zero.     Solve   these 
equations  simultaneously. 

28.  The  same   result  will  be    obtained  if   we   begin   by 
multiplying  each  observation  equation  by  the  square  root  of 
its   weight   and   then   proceed   according   to   the   first   rule 
(paragraph  25). 

This  result  illustrates  the  important  principle  that  multi- 
plying a  set  of  equations  by  the  square  roots  of  their  weights 
reduces  them  all  to  equivalent  equations  of  weight  unity. 

29.  Relation  between  the  Weight  of  an  Observation  and 
the  Value  of  h.     If  in  paragraph  20   the    n   observations 


THE  ADJUSTMENT  OF  OBSERVATIONS.  21 

have   weights  jt>x,  jt>2,  .  .  .  pn,    and   the   quantity    A   values 
AU  A2,  .  .  .  An,  then  equation  (18)  becomes 

P=kle~h^v^  Jc2e-h**vf  .  .  .  kne~h^v^  (dv)n 


The  most  probable  set  of  values  of  the  unknowns  is  that 
which  makes  P  a  maximum,  and  P  is  a  maximum  when 

li\  Vi'2  +  ^22  ^22+  •  •  •  hrfvn2  is  a  minirmim.   (23) 

The  conditions  for  a  minimum  value  of  this  expression  are 
the  following,  which  are  then  for  this  case  the 

NORMAL    EQUATIONS. 


But  equations  (21)  are  also  the  normal  equations  for 
this  case.  Hence  (21)  and  (24)  must  be  identical,  and 

pi  :  Pz  :   •      Pn  =  /*i3  :  hf  :   .  .  .  h,?        (25) 

Tliat  is,  the  square  of  A  is  proportional  to  the  weight  of 
the  observation.  Accordingly,  since  A  increases  in  value  as 
the  quality  of  the  observations  is  improved,  it  is  culled  "  The 
Measure  of  Precision." 


22  METHOD   OF  LEAST  SQUARES. 

Further,  it  follows  from  (23)  and  (25)  that  the  most 
probable  system  of  values  of  the  unknowns  will  be  that  in 
which 

Pi  Vi*-\-paVt*-{-  •  •  -Pn^n    is  a  minimum.    (26) 

And  this  is  the  most  general  form  of  statement  of  the 
principle  of  "  Least  Squares."  The  same  principle  is  repre- 
sented in  equations  (21). 

30.  Computation  of  Corrections.    If  large  numbers  occur 
in  the  observations  it  is  better  to  compute  the  most  probable 
corrections  to  apply  to  the  observed  values  rather  than   the 
most  probable  values  of  the  unknowns  themselves.     In  this 
way  we  can  often  avoid  a  large  amount  of  numerical  work. 

31.  Example.     P^  jP2,  jP8,  P4,  P&   are  five  points  whose 
altitudes  above  the  mean  level  of  the  sea  are  to  be  determined 
from  the  following  observations  of  difference  of  level. 

Px  =  573.08  P4-P2  =  170.28 

Py-Pl  =  2.60  Pt—P,=  425.00 

P2  =  575.27  P6  =  319.91 

PS—P2  =  167.33  P6  =  319.75 

P4-P8  =  3.80 

An  inspection  of  these  observations  shows  that  we  may  put 
^  =  573-1-2!  P4=745  +  24 


(A) 

P8=742  +  23 

where    zl5  22,  2S,  24,  26,    are    small    corrections    whose   most 
probable  values  are  to  be  determined.     We  now  have  for 

OBSERVATION    EQUATIONS. 
573  _|_  Zl  _  573.08  =  0       or  2!-.  08  =  0 

_  573_2l_       2.60=0       or       22-2!-.  60  =  0 
575  -f  22  -  575.27  =  0       or  22  —  .27  =ty 


THE  ADJUSTMENT  OF  OBSERVATIONS.  23 

742  +  z8  -  575  -z2-  167.33  =  0  or  z8-z2-.33  =  0 

745  _|_  24  —  742  -  z3  —      3.80=0  or  z4-z3— .80=0 

745  +  z4— 575  — za- 170.28=0  or  24-z2-.28=0 

745  _|_  24  _  320  -z6-  425.00=0  or  z4  —  z5            =0 

320  +  26-  319.91  =  0  or  z6  +  .09  =  0 

320  -i-z6-  319.75=0  or  z6-f.25  =  0 

From  these  we  now  form  the 

NORMAL    EQUATIONS. 

22X-    z2  +   .52=0 

_    2l_|_422_    zs—    z4  —    .26=0 

-    z2-f2z8-.    z4  -j-    .47  =  0 

_    22_    28-|-3z4  —    z5— 1.08=0 

-    z4-f-3z5-f-    .34  =  0 

and  solving, 

2l=-.19;    z2  =  .14;    zs=.05;    z4=.43;    z5=.03 

Substituting  these  in  equations  (A)  we  have  for  the  most 
probable  altitudes, 

Pl  =  572.81          Ps=  742.05          P5  =  320.03 
P2  =  575.14          P4  =  745.43 

If  the  original  observation  equations  had  been  retained, 
the  independent  terms  in  the  normal  equations  would  have 
been 

570.48         240.26         163.53         599.08         214.66 

32.  Significant  Figures.  The  adjustment  by  the  Method 
of  Least  Squares  of  observations  which  occur  in  practice, 
although  not  difficult,  is  apt  to  be  long  and  laborious.  Hence 
to  reduce  this  labor  as  much  as  possible  it  is  of  great  import- 
ance that  careful  attention  should  be  given  in  the  solutions 
to  the  proper  use  of  significant  figures.  When  in  doubt, 


24  METHOD   OF  LEAST  SQUARES. 

however,  as  to  the  proper  number  of  figures  to  retain  it  is 
better  to  keep  too  many  rather  than  too  few,  as  the  superfluous 
figures  can  be  rejected  at  the  end  of  the  computation ;  while 
if  too  few  are  retained  the  results  obtained  from  the  compu- 
tations will  be  worthless. 

For  a  general  discussion  of  the  subject  of  significant  figures 
see  Holman's  "  Precision  of  Measurements,"  pages  76  to  84, 
but  for  the  present  the  following  rules  will  suffice  for  most 
cases. 

Rule  1.  In  casting  off  places  of  figures  increase  by  1  the 
last  figure  retained,  when  the  following  figure  is  5  or  over. 

Rule  2.  In  the  precision  measure  retain  two  significant 
figures. 

Rule  3.  In  any  quantity  retain  enough  significant  figures 
to  include  the  place  in  which  the  second  significant  figure 
of  its  precision  measure  occurs. 

Rule  4.  When  several  quantities  are  to  be  added  or 
subtracted,  apply  Mule  3  to  the  least  precise  and  keep  only 
the  corresponding  figures  in  the  other  quantities. 

Rule  5.  When  several  quantities  are  to  be  multiplied  or 
divided  into  each  other,  find  the  percentage  precision  of  the 
least  precise.  If  this  is 

1  per  cent  or  more,  use  four  significant  figures. 
.1     "        "      «       "         "    five  "  " 

.01  "        "      "       "         «    six  "  " 

in  all  the  work.  If  the  final  result  obtained  in  this  way 
conflicts  with  Rule  3,  apply  the  latter. 

Rule  6.  When  logarithms  are  used,  retain  as  many 
places  in  the  mantissce  as  there  are  significant  figures 
retained  in  the  data  under  Rule  5. 

The  application  of  these  rules  is  not  always  possible  in  the 
course  of  the  work,  since  the  precision  measures  may  not  be 
known  until  the  end  of  the  computation.  But  as  a  general 
rule  it  is  sufficient  in  direct  observations  to  retain  one  more 


THE  ADJUSTMENT  OF  OBSERVATIONS.  25 

place  of  figures  than  is  given  by  the  individual  observations, 
and  in  indirect  observations  to  retain  two  additional  places. 

CONDITIONED  OBSERVATIONS. 

33.  Conditioned  Observations  are  those  in  which  the 
unknown  quantities  must  be  determined  not  only  so  as  to 
satisfy  as  closely  as  possible  the  observation  equations,  but 
also  so  as  to  satisfy  exactly  certain  other  conditions.  These 
conditions  must  be  less  in  number  than  the  unknown  quantities, 
otherwise  the  unknowns  could  be  determined  from  the  con- 
ditions alone. 

The  adjustment  of  observations  of  this  class  may  be  reduced 
to  the  method  already  used  for  unconditioned  observations  in 
the  following  way. 

The  observations  are  represented  by  "  Observation  Equa- 
tions," and  the  conditions  by  certain  other  equations,  called 
"  Condition  Equations." 

Between  these  two  sets  of  equations  we  will  eliminate  as 
many  unknowns  as  there  are  conditions.  From  the  resulting 
equations,  which  will  be  the  same  in  number  as  the  observa- 
tions, we  will  form  in  the  usual  manner  the  "  Normal 
Equations"  for  the  remaining  unknowns.  Having  solved 
these  normal  equations  and  substituted  the  results  in  the  con- 
dition equations,  we  shall  obtain  the  values  of  the  unknowns 
first  eliminated. 

All  conditions  of  the  problem  are  now  fulfilled,  for  the 
condition  equations  are  satisfied  exactly  and,  moreover, 
according  to  the  principle  of  Least  Squares  our  results 
are  those  rendered  most  probable  by  the  existence  of  the 
observations. 

As  in  the  example  last  considered,  it  is  often  more 
advantageous  to  compute  corrections  to  the  observed  values 
of  the  unknown  quantities  rather  than  the  values  of  the 
quantities  themselves. 


26  METHOD   OF  LEAST  SQUARED. 

34.    Example.    Find  the  most  probable  values  of  the  angles 
of  a  quadrilateral  from  the  observations, 

^  =  101°  13'  22"  weight  3 

J5  =   93    49   17        "        2  .. 

O=   87      5   39        «        2 

D=   77    52   40        «        1 


0'  58" 
The  condition  to  be  satisfied  is  in  this  problem 

(B) 


Let  2u  z2,  28,  z4   be  the  most  probable  corrections  to  add 
to  the  observed  values.     This  gives  for 

OBSERVATION  EQUATIONS 
zl=Q  weight  3 

*;=o  "  2  <C) 

and  for  the     • 

CONDITION    EQUATION 

Eliminating  z^  between  (D)  and  (C),  the  equations  from 
which  the  normal  equations  are  to  be  derived  become 


=  0  weight  3 

(E) 


z2  =  0        "       2 
a,  =  0        "       2 


THE  ADJUSTMENT  OF  OBSERVATIONS.  27 

Applying  the  rule  in  paragraph  27,  these  give  the 


NORMAL    EQUATIONS 


(F) 


Solving,  and  substituting  the  results  in  equation  (D),  we 
find 

zl=-    8.29  e,=  -  12.43  Q. 

22  =  -  12.43  s4  =  —  24.85 

Applying  these  corrections  to  the  observations  (A),  the 
most  probable  values  of  the  angles  are 

-4  =  101°  13'  13".71 

.#=93  49        4.57                           H, 

C=    87  5  26.57 

D=    77  52  15.15 

Note.  In  eliminating  unknowns  between  the  observation 
and  condition  equations  care  must  be  taken  that  the  obser- 
vation equations  are  not  combined  with  each  other  or  multi- 
plied by  any  quantity.  For  if  this  is  done  the  weights  of 
the  observation  equations  will  be  altered.  (See  §28.) 

35.  In  the  above  example  it  is  evident  that  the  corrections 
to  be  applied  to  the  different  observations  are  inversely  as 
their  weights.  And,  in  general,  when  there  is  but  one 
equation  of  condition,  the  observations  expressing  direct 
determinations  of  the  unknowns,  the  corrections  will  be  pro- 
portional to  the  coefficients  of  the  unknowns  in  the  equation 
of  condition  divided  by  the  weights  of  the  corresponding 
observations.  A  proof  of  this  is  given  in  paragraph  117. 


28  METHOD   OF  LEAST  SQUARES. 

The  most  common  case  is  that  in  which  these  coefficients 
are  all  unity,  as  in  the  example  just  solved,  and  we  may  then 
derive  the 

Rule.  Find  the  difference  between  the  theoretical  and 
observed  results  and  divide  this  correction  among  the 
observations  in  the  inverse  ratio  of  their  weights. 

In  the  last  example  the  sum  of  the  observed  angles  exceeds 
360°  by  58".  Therefore  the  correction  to  be  applied  to  A  is 


-  58  X  -  -=-58xi=-  8.29 


EMPIRICAL   FORMULAS  AND  CONSTANTS. 

36.  In  the  work  so  far  considered  the  observations  are 
supposed  to  be  made  either  directly  upon  the  values  of  the 
unknown  quantities  or  upon  some  function  of  the  unknowns 
whose  form,  and  the  constants  entering  into  it,  are  definitely 
known.  But  another  sort  of  problem  frequently  occurs, 
in  which  observations  are  made  upon  the  values  of  a  certain 
variable  and  the  corresponding  values  of  some  function  of  it, 
the  exact  form  of  the  function  not  being  known.  The  object 
in  this  case  is  the  determination  of  the  most  probable  form  of 
the  function  and  the  values  of  the  constants  involved  ;  that 
is,  the  derivation  of  the  algebraic  expression  best  representing 
the  law  connecting  the  variable  and  function. 

This  expression  may  be  looked  upon  as  the  equation  of  a 
curve,  abscissas  denoting  values  of  the  variable  and  ordinates 
values  of  the  function,  and  for  all  values  of  the  variable 
within  the  range  of  the  observations  we  may  determine  from 
it  the  most  probable  values  of  the  function  corresponding. 
But  except  in  special  cases,  where  the  number  of  observations 
is  large,  where  the  law  connecting  variable  and  function  is 
well  defined,  and  where  the  equation  obtained  is  an  accurate 


THE  ADJUSTMENT  OF  OBSERVATIONS.  29 

representation  of  this  law,  it  cannot  be  assumed  to  apply 
beyond  the  range  of  the  observations.  And  in  no  case  would 
it  be  safe  to  make  use  of  the  curve  very  far  beyond  the  limits 
of  the  observations. 

37.  The  Method  of  Least  Squares  will  not  assist  in  deter-  M, 
mining  the  form  of  the  function.     This  must  be  settled  upon 
beforehand,   either   from   theoretical    considerations    or    by 
constructing  a  plot,  using  values  of  the  variable  as  abscissas 
and  of   the  function  as  ordinates,  when  the  smooth  curve 
drawn  through  the  points  thus  obtained  will  indicate  the 
form  of  equation  to  be  used. 

It  is  to  be  observed  that  this  is  a  method  of  trial  and  will 
not  necessarily  give  the  most  probable  form  of  the  function  ; 
and  in  fact  we  may  not  be  able  to  obtain  the  form  that  would 
be  absolutely  best.  Further,  several  forms  of  equation  may 
be  known  which  would  represent  well  the  plotted  points.  In 
such  a  case  that  should  be  considered  the  best  in  which  the 
sum  of  the  squares  of  the  residuals  is  found  to  be  the  least. 

38.  As  soon  as  the  form  of  the  function  is  decided  upon  it 
should  be  reduced  to  the  linear  form,  and  the  determination 
of   the  values  of   the  constants  involved  is  then  a  simple 
application  of  the  preceding  methods. 

As  the  "  Observation  Equations  "  in  any  given  problem  will 
all  be  of  the  same  kind,  it  is  usually  advisable  to  write  out 
the  general  form  of  the  "Normal  Equations"  and  arrange 
the  computations  in  tabular  form,  while  the  retention  of 
the  proper  number  of  significant  figures  is  of  particular 
importance  in  this  work. 

39.  A  case  that  frequently  occurs  is  that  in  which  the 
quantity    y   is  a  constantly  increasing  function  of  the  vari- 
able a;,  or  where  the  plotted  curve  is  approximately  parabolic 
in  form.     Here  the  equation 


...  (27) 

may  be  taken  to  represent  the  relation  between  the  variable 


30 


METHOD  OF  LEAST  SQUARES. 


and  the  function.  The  larger  the  number  of  terms  taken  in 
the  second  member,  the  more  accurately  may  the  equation 
obtained  be  made  to  represent  the  results  of  the  observations  ; 
but  the  labor  involved  increases  rapidly  with  increase  in  the 
number  of  terms,  and  if  the  plot  shows  a  very  nearly  straight 
line  the  first  two  terms  alone  may  suffice. 

40.    Example.     In  measuring  the  velocity  of  the  current 
of  a  river  the  following  results  were  obtained  : — • 

Depths.  Velocities. 

x  V 

1  4.86 

2  5.14 

3  5.15 

4  4.85 

5  4.24 

6  3.36 

7  2.16 

8  0.67 

The  velocity  at  the  surface  is  4.250.     Find  the  equation  of 
a  curve  which  will  express  the  relation  between  x  and  V. 
Plotting  the  observations  we  find  the  curve 


ADJUSTMENT  OF  OBSERVATIONS, 


This  is  approximately  parabolic  in  form  and  passes  through 
the  fixed  point  (0,  4.25).  Therefore  the  relation  between  x 
and  V  may  be  expressed  by  the  equation 


(A) 


and  substituting  in  this  the  corresponding  values  of  x  and  V 
as  given  by  the  observations,  we  shall  have  eight  observation 
equations  from  which  the  most  probable  values  of  B  and  (7 
are  to  be  computed. 

All  of  the  observation  equations  being  of  the  form  (A)  we 
have  the 


NORMAL    EQUATIONS 


4  4-  B  Sx8  +  4.25  2z2  -  2  Fa;2  =  0 


(a) 
(b) 


For  computing  the  coefficients  in  these  equations  it  is  most 
convenient  to  arrange  the  following  table. 


X 

V 

Vx 

X* 

Fa;2 

x8 

x* 

1 

4.86 

4.86 

1 

4.86 

1 

1 

2 

5.14 

10.28 

4 

20.56 

8 

16 

3 

5.15 

15.45 

9 

46.35 

27 

81 

4 

4.85 

19.40 

16 

77.60 

64 

256 

5 

4.24 

21.20 

25 

106.00 

125 

625 

6 

3.36 

20.16 

36 

120.96 

216 

1296 

7 

2.16 

15.12 

49 

105.84 

343 

2401 

8 

0.67 

5.36 

64 

42.88 

512 

4096 

86 

111.83 

204 

525.05 

1296 

8772 

2* 

SFse 

2-e2 

2Fz2 

2*» 

2*4 

32  METHOD   OF  LEAST  SQUARES. 

Substituting  these  results  in  (a)  and  (b)  we  have 

8772  6^-1-1296^  -f  341.95  =  0  (c) 

1296(7+    204^-f-    41.17  =  0  (d) 

and  solving, 

C  =—.1493  J?  =  .7465  (e) 

Therefore  the  required  equation  is 

V  =  4.25  -f-  .7465z  -  .1493<c2  (f ) 

Whenever  the  quantities  in  the  observations  are  so  large 
that  the  use  of  logarithms  is  desirable,  these  can  best  be  put 
in  the  same  columns  directly  over  the  natural  numbers. 

41.  It  is  to  be  remarked  that  in  solutions  like  the  above 
we  assume,  from  our  method  of  forming  the  observation  and 
normal  equations,  that  the  observations  on  the  values  of  the 
function  are  alone  subject  to  error,  the  observations  on  the 
variable  being  supposed  to  be  exact  or  to  have  errors  so  small 
as  to  be  negligible. 

42.  Periodic  Phenomena.    If  as  the  variable  increases  the 
function  passes  through  recurring  values,  that  is,  if   y   is   a 
periodic  function  of   x,  some  form  of  trigonometric  equation 
would  be  the  proper  one  to  select.     For  instance,  a  good  form 
to  use  is 

y  =  A  +  B  sin  ^jfv+  C  cos^jfx          (28) 

where  A,  .Z?,  C  are  the  constants  whose  most  probable  values 
are  to  be  found,  and  m  is  the  number  of  units  of  x  comprised 
in  the  entire  cycle  of  values  of  y.  The  quantity  m  is  to  be 
determined  from  an  inspection  of  the  observations,  and  if  it 
appears  that  several  values  of  m  might  be  used,  all  should  be 
tried,  and  that  which  leads  to  the  smallest  sum  for  the  squares 
of  the  residuals  is  to  be  considered  the  best.  If  the  several 


THE  ADJUSTMENT  OF  OBSERVATIONS.  33 

cycles  are  not  similar  and  regular,  additional  terms  involving 
multiples  of  x  will  have  to  be  added  to  equation  (28). 

43.  Special  Treatment  of  Exponential  Equations.  If  the 
equation  selected  is  not  in  the  linear  form  as  regards  the 
unknown  constants  the  general  method  of  procedure  is 
given  in  paragraph  44,  but  in  some  special  cases  a  more 
simple  reduction  is  possible.  A  quite  common  case  is  the 
following:  — 

Suppose  the  relation  between  x  and  y  is  expressed  by  the 
equation 

y  =  kxm  (29) 


the  problem  being  to  obtain  the  best  values  of  k  and  m. 
Taking  the  logarithms  of  both  members  of  (29), 

log  y  =  log  k  -j-  m  log  x 

Denoting  log  k  by  k',  this  equation  becomes 
m  log  x-\-k'  —  log  y  =  0 

which  is  in  the  linear  form  as  regards  the  unknowns  m  and 
&',  and  the  normal  equations  may  now  be  formed  in  the  usual 
manner. 

The  most  convenient  way  of  determining  whether  equation 
(29)  is  a  suitable  one  to  select  or  not  is  to  plot  the  corre- 
sponding values  of  x  and  y  on  logarithmic  cross-section 
paper.  If  (29)  is  a  proper  equation  to  use  the  plotted  points 
will  lie  upon  a  straight  line. 

REDUCTION  OF  OBSERVATION  EQUATIONS  TO  THE 
LINEAR  FORM. 

44.  When  the  observation  equations  are  not  linear  as 
regards  the  unknowns,  the  only  practicable  method  of  pro- 
cedure, as  already  mentioned  in  paragraph  23,  is  to  reduce 


34  METHOD  OF  LEAST  SQUARES. 

them   to   that   form.     This   reduction    may  be    effected    as 
follows  :  — 

Let  the  observation  equations  be 


Z2,  .  .  .  Zq)  =  M[ 
Z2,  .  .  .  Zq)  =  MS 

.....  (A) 

fn(Zl  Z2,  .  .  .  Zq)  =  Mn 

in  which  Zx,  Z2,  .  .  .  Zg  represent  the  unknown  quantities  and 
J/i,  J/g,  .  .  .  J/n  the  observations,  the  functions  being  of 
known  form. 

Let  Zt',  Z2',  .  .  .  Zq  be  approximate  values  of  Zt,  Z2,  .  .  .  Z9, 
found  by  trial  or  the  solution  of  a  sufficient  number  of  the 
observation  equations,  and  let  the  most  probable  values  of 
ZM  Z2,  .  .  .  Zq  be 

^'  +  2!,  Z2'  +  22,  ...Zg'  +  z9  (B) 

«!,  z2,  .  .  .  3?  being  small  corrections  whose  values  are  to  be 
determined  by  the  Method  of  Least  Squares.  The  first 
observation  equation  in  (A)  may  then  be  written, 


Expanding  the  first  member  by  Taylor's  Theorem,  denoting 
^,  Z2',  .  .  .  Zq)  by  A?j,  and  neglecting  terms  containing 
powers  of  2t,  z2,  .  .  .  zq  higher  than  the  first,  this  becomes 


If  now  we  represent  the  coefficients  of   2,,  2,,  ...  zgj   by 
aii  ^i»  •  •  •  ^u  a^8°    ^i  —  -^i    ^y   Wiu    an<^    treat   the    other 


THE  ADJUSTMENT  OF  OBSERVATIONS.  35 

equations  in  (A)  in  the  same  manner,  our  observation  equa- 
tions will  take  the  form 


(C) 


The  second  members  reduce  to  the  residuals  since  the 
quantities  zt,  22,  •  •  •  zq  represent  merely  the  most  probable 
values  of  the  corrections.  Equations  (C)  can  now  be  solved 
in  the  usual  manner  and  the  most  probable  values  of 
Zj,  Z2,  .  .  .  Zq  found  by  substituting  the  results  in  (B). 

45.  Example.  From  the  following  observations  find  the 
most  probable  values  of  x  and  y. 

sin  x  -4-  cos  2y  =  1.5 
cos  x  -\-  3  sin  y  =  1.7  (A) 

x2  4  5y  =  2.1 

By  trial  it  is  found  that  42°  and  18°  are  approximate 
values  of  x  and  y.  Then  in  accordance  with  paragraph  44 
we  put 

x  =  42°4-z1  y=18°+2a  (B) 

and  expanding  the  different  functions,  we  find 

&!  =  sin  42°  4-  cos  36°,      ^-l  =  cos  42°,      -.- -  =  -  2  sin  36° 

ox  vy 

=  1.48  =.74  =-1.18 

£2  =  cos  42°  4-  3  sin  18°,    ~-2  =  -sin42°,    vr-a  =  3  cos  18° 

ox  vy 

=  1.67  =-.67  =2.85 

*-/42irV4-618ir          **-_o42*          dk» 

n.«  I   — r—  O  .  'z —  —  it  .  7; — • 

\1KO/  180  °x  180          dy 

=  2.11  =1.47 

Art  -  Jl/i  =  -  .02,          A-2  -  J/a  =  -  .03,          *,  —  J/3  =  .01 


86  METHOD    OF  LEAST  SQUARES. 

Therefore  we  have  for 

OBSERVATION    EQUATIONS 

.742!  —  1.18za  —  .02  =  vl 
-  .672X  -f  2.8522  —  .03  =  u2  (C) 

1.472l  _j_       5z2  _|_  .01  =  va 

From  these  are  obtained  the 

NORMAL    EQUATIONS 

3.162!  -L.    4.5722  +  .020  =  0  D. 

4.572!  -[-  34.5l22  —  .012  =  0 

Solving, 

Zi  =  -  .00845  22  =  .00147  (E) 

These  results  are  in  circular  measure.  Reducing  to  degree 
measure  we  have 

a1==   -29'.0  22  =  5'.1  (F) 

Substituting  these  in  (B), 

a;:=410  Sl'.O  y=18°5'.l  (G) 

46.  If  the  observations  are  conditioned,  precisely  the  same 
method  will  be  followed  in  reducing  all  the  equations  to  the 
linear  form.  The  rest  of  the  solution  will  then  be  as  usual. 

If  the  values  found  for  2j,  22,  .  .  .  zq  should  turn  out  to  be 
so  large  that  the  terms  involving  their  second  and  higher 
powers  can  not  be  neglected  as  assumed,  the  process  must  be 
repeated  using  the  values  of  Z^  Z2,  .  .  .  Zq  first  obtained  as 
approximations. 

In  a  few  cases  the  equations  may  be  reduced  to  the  linear 
form  by  some  special  artifice  of  a  simple  character,  as  in 
paragraph  43.  In  such  cases  the  method  of  expansion  by 
Taylor's  Theorem  should  not  be  used. 


J 


CHAPTER  III. 

THE   PRECISION   OF   OBSERVATIONS. 


47.  The  work  so  far  considered  has  treated  solely  of  the 
methods  by  means  of  which  the  most  probable  values  of  the 
unknown    quantities   may  be   determined  from  a  series  of 
observations.     But  in  general  something  more  than  this  is 
desired.     We  wish  to  know,  if  possible,  how  much  reliance 
can  be  placed  upon  the  results  obtained,  and  how  they  com- 
pare in  precision  with   other   determinations  of   the   same 
quantities.     Preliminary  to  the  discussion  of  the  precision  of 
our  results  it  will  be  necessary  to  consider  more  fully  the 
Curve  of  Error. 

48.  The  Constant  k.     The  law  of  distribution  of  errors  of 
observation  has  been  shown  to  be  represented  by  a  curve 
whose  equation  is 

y  =  ke-™ 

In  this,  if  x  =  0,      y  =  k  (30) 

Therefore  the  constant  k  represents  the  intercept  of  the 
Curve  of  Error  on  the  axis  of  Y.  It  is  not,  however,  an 
independent  quantity  but  is  determined  by  the  value  of  A, 
as  will  now  be  shown. 

49.  To  Find  k  in  Terms  of  h.     Since,  as  shown  in  para- 
graph   18,  the    total    area   between  the  curve  and   the  axis 
of    A"  is  denoted  by  unity,  we  have 


k  C"e-h'2x*dx  =  1     or     k  C^e^^dx  =  5          (a) 

J-ao  Jo 

This  may  be  written 

/*<*      ^ 

I       g~h' 
Jo 


38  METHOD   OF  LEAST  SQUARES. 

Let   t  =  hx,    .-.    dt  =  hdx.     Also,  when    x  =  oc,  £  =  cc, 
and  when    x  =  0,   Z  =  0. 


^a52A  <?«  (c) 

o 
Multiplying  this  equation  by 

/%«  /»00 

I      e-*2^  =   I 

•/  0  ^  0 


we  have 


r  r  **  dt  T  =  r 

|_*^o  J          */o    »^o 


_  1  /••    <7» 

~  2  Jo    1-fa;2 

1f~~  ~~.   00 

i  T 

=  —     tan  J  a;        =  — 
2L  Jo         ^ 


Substituting  (31)   in   (c)  and  (b) 

\fc  h  h 


2  2 

Therefore  equations  (16)  and  (17)  become 


~ 

y  —  ~  e 

VTT 


or  =  —  -  (32) 


(33) 


THE  PRECISION  OF  OBSERVATIONS.  39 

50.  It  has  been  shown  that  the  quantity  A    is  a  "Measure 
of  Precision"  of  the  observations,  and  hence  a  determination 
of  its  value  in  each  case  would  enable  us  to  compare   the 
relative  reliability  of  different  measurements.      In   practice, 
however,  it  is  not  found  convenient  to  compute  the  value  of 
h   directly,  and  so  this  quantity  is  used  only  in  developing 
the  theory  of  the  subject,  while  in  the  comparison  of  observa- 
tions certain  other  quantities  now  to  be  derived  are  used  as 
precision  measures.     These  latter  quantities  are  called  the 
"Mean  of  the  Errors,"  the  "Mean  Error,"  and  the  "Probable 
Error,"  respectively,  and  may  be  computed  directly  from  the 
observations.     Further,  as  it  will  be  found  that  they  all  bear 
a  definite  relation  to   A,   the  value  of  this  quantity  can  be 
determined  from  them  if  desired. 

In  the  following  discussions  no  distinction  is  made  between 
positive  and  negative  errors  of  the  same  numerical  magnitude, 
and  unless  otherwise  stated  the  observations  are  all  of  the 
same  weight. 

THE  MEAN  OF  THE  ERRORS  OR  AVERAGE  DEVIATION. 

51.  The  Average  Deviation   (a.d.)   of  an  observation  is 
the  arithmetical  mean  of  the  errors  all  taken  with  the  positive 
sign. 

Since  from  (6)  the  probability  that  the  error  of  a  single 
observation  will  fall  between  x  and  x-\-dx  is 

<£(«)  dx, 

if   n   observations  are  made,   the  number  of  errors  falling 
between  these  limits  is 

n  <f>(x)  dx. 
Hence  the  sum  of  all  the  errors  of  the  observations  is 

,%<*>  x»QO 

n  I      y  <j>(x)  dx      or       2w  I      x  <f>(x)  dx. 

*/-»  Jt 


40  METHOD   OF  LEAST  SQUARES. 

Dividing  this  last  expression  through  by    n    we  have 

/*°D  2A  /•* 

a.d.  =  2  I     x<f>(x)dx    =    —I     <rA2a;2a:<fo 

Jo  VV^o 

=  -  -1-  f  Y*2*2  (-2A2z)  rfz 

AVTT^O 

or  «.<?.   =  -  (34) 


THE  MEAN   ERROR. 

52.  The  Mean  Error  (p)  of  an  observation  is  the  square 
root  of  the  arithmetical  mean  of  the  squares  of  the  errors. 

The  total  number  of  errors  being  n,  the  number  falling 
between  x  and  x  -\-  dx  is,  as  just  shown, 

n  <f>(x)  dx, 
and  the  sum  of  the  squares  of  these  errors  is 

n  x2  <f>(x)  dx. 
Therefore  the  sum  of  the  squares  of  all  the  errors  is 

n  I      x2  <f>(x)  dx 
•£_« 

Ha  =  -  -   C     e-h°-x*  X2  jx  (%) 

But  as  shown  in  paragraph  49, 
h 


dx  =  I       or  e-l™  dx  =  If        (b) 

VTT^-"  ^-«  h 

Differentiating  (b)  with  respect  to    A, 
-2A 


_00 


THE  PRECISION  OF  OBSERVATIONS. 


41 


and  replacing  the  integral  in  (a)  by  its  value  as  determined 
from  equation  (c),  we  have 


THE  PROBABLE  ERROR. 

53.  The  Probable  Error  (r)  of  an  observation  is  an  error 
such  that  one-half  the  errors  of  the  series  are  greater  than  it 
and  the  other  half  less  than  it.  Or  it  is  an  error  of  such  a 
magnitude  that  the  probability  of  making  an  error  greater 
than  it  in  any  given  observation  is  just  equal  to  the  probability 
of  making  one  less  than  it,  both  probabilities  being  one-half. 

The  probability  that  the  error  of  an  observation  will  fall 
between  x  and  x-\-dx  being  <f>(x)  dx,  the  probability 
that  the  error  will  fall  between  the  limits  r  and  —  r  is 


P  = 


(36) 


If   r   is  the  probable  error,  P   is  one-half,  or 


(37) 


and  from  this  definite  integral   r   is  to  be  found. 

Let  t  =  A«,  .-.  dt  =  hdx.  Also  when  x  =  r,  we  have 
t  =  hr,  and  when  x  =  0,  t  =  0.  Substituting  these  results 
in  (37),  we  have 


VTT 


(38) 


42  METHOD   OF  LEAST  SQUARES. 

Denote  hr  by  p.  Then  by  interpolation  in  a  table  of 
values  of  this  integral,  the  value  of  hr  in  (38)  is  found  to 
be 

p  =  hr  =  .47694 

._£_..  .17691 
=  ft  :         h 

54.    When   Z  is  small  the   values   of    J 

Jo 
found  by  expanding  e~^  into  a  series  and  integrating   the 

successive  terms.     Thus,  by  Maclaurin's  Theorem, 


fV* 

Jo 


*8  /6  *7 

-        + 


3         5[2_        7[3_ 

RELATIONS  BETWEEN    Jl,   r,   «.d.,   A,    AND    p. 

P  1 

.    From  (40)    r  =  -,  and  from  (34)   a.c?.  =  — 

h 


55 


r  =  p  a.d.  V/TT 
or  r  =  .8453  a.<?.  (41) 

Also  from  (35)    p.  = 


or  r  =  .6745  |i  (42) 


THE  PEECISION  OF  OBSEBVATIONS.  43 

The  relation  between  the  values  of  /*,  r,  a.d.,  and  h  may 
be  conveniently  expressed  as  follows  :  — 

1          r 
o   =  -  =  -  =   a.d.  y/?  (43) 


or  arranging  in  tabular  form, 


f- 

r 

a.d. 

/* 

1.0000 

1.4826 

1.2533 

r 

0.6745 

1.0000 

0.8453 

a.d. 

0.7979 

1.1829 

1.0000 

From  this  it  will  be  seen  that 

/i  >  a.d.  >  r  (44) 

56.    Further,  since  in    (25)   it  was  shown  that  p  oc  A2,    it 
follows  at  once  from  (43)   that 


*    x  h'      (45> 


That  is,  the  weights  of  different  determinations  of  a 
quantity  vary  inversely  as  the  squares  of  their  Mean  Errors, 
their  Probable  Errors,  or  their  Average  Deviations. 

It  is  to  be  observed,  however,  that  the  determination  of  the 
relative  weights  of  quantities  from  a  comparison  of  their 
precision  measures  according  to  (45)  applies  only  when  the 
quantities  are  of  the  same  kind  and  subject  to  the  same  con- 
stant errors,  if  any  of  the  latter  exist.  (See  §  98.)  The 
applications  of  (45)  are  numerous  and  important. 

57.  Example  A.  Suppose  n  direct  observations,  all  of 
the  same  weight,  be  made  upon  a  quantity,  and  that  the 
probable  error  of  a  single  observation  is  r.  Then  since  the 


44  METHOD  OF  LEAST  SQUARES. 

weight  of  the  arithmetical  mean  is    n,    its  probable  error    r0 
will  be  given  by 

r 2          1  r 

-~  =  -  or         r0  =  —=  (46) 

Or  in  general,  suppose  8  is  any  precision  measure  of  an 
observation  of  weight  p,  and  suppose  p0  is  the  weight  of  a 
second  similar  quantity  or  observation,  then  the  correspond- 
ing precision  measure  80  of  the  latter  will  be 

(47) 

The  case  of  most  common  occurrence  is  that  in  which 
p  =.  1,  and  then  we  have 

80  =  -^=  (48) 


Example  J5.  A  line  is  measured  five  times  and  the  aver- 
age deviation  of  the  mean  (A.  D.)  found  to  be  .016  feet. 
How  many  additional  measurements  are  necessary  in  order 
that  the  A.  D.  of  the  mean  may  be  reduced  to  .004  feet  ? 

Let  x  be  the  total  number  of  observations  required.    Then 

x   :    5  =  .000256    :    .000016 

x  =  80 

Consequently  the  number  of  additional  measurements 
required  is  75. 

Example  C.  In  two  determinations  of  the  quantity  L 
there  were  obtained 

Lv  =  427.320  ±  0.040,         Z2  —  427.30  ±  0.16 

Find  their  relative  weights,  and  the  most  probable  value 
of  L  and  its  probable  error. 


THE  PEECISION  OF  OBSERVATIONS.  45 

Note.  The  above  is  the  method  commonly  employed  to 
denote  that  the  probable  errors  of  the  observations  are  0.040 
and  0.16. 

»,          162          16 
From    (45)  — *  =  -   -  =    - 

^2          42  1 

From    (3),  the  most  probable  value  of   L    will  be  given  by 
'    ^  =  427   +   16  X  .32  +  .80 

=  427.319 
and  the  weight  of   L0   being    17,   by    (48), 

r0  =   -^—  =  .039 

vrf 

Therefore  we  should  write  the  result 

Z0  =  427.319  ±  .039 


REPRESENTATION  OF  [1,  a.d.,  AND  r  ON  THE  CURVE  OF 

ERROR. 

58.    To  find  the  points  of  inflection  of  the  Curve  of  Error 
we  have 


.- 
ax 


y  =  ke-- 
-  2  h*k  x 


d*y 

For  a  point  of  inflection,      ,  '2    =    0,    or 

2  A2  k  e~h-x'2  (2  A*  x2  -  1  )  =  0 

x  =         =  ^     by  (35). 
Ay  2 


46  METHOD    OF  LEAST  SQUARES. 

That  is,  the  Mean  Error  is  represented  by  the  abscissa  of 
the  point  of  inflection  of  the  Curve  of  Error.  See  OM  in 
figure  2,  page  9. 

Next,  for  the  abscissa  of  the  centre  of  gravity  of  the  area 
to  the  right  of  OY,  we  have 

I     y  x  dx 

aso  =  OD  =  Jo 

i\i     /-/ rp 

y  ' '  Ju 

2  A 


/,a 
=  7=    f 

\ir     J o 


for,  (§  18),  j^y  dx  =  ^ 

Integrating, 

1 

=    a.d.  by  (34). 


Finally,  if  an  ordinate  PP'  be  drawn  so  as  to  bisect  the 
area  to  the  right  of  the  origin  between  the  curve  and  the 
axis  of  X,  the  Probable  Error  will  be  represented  by  the  dis- 
tance of  this  ordinate  from  the  axis  of  Y,  For  the  proba- 
bility of  the  occurrence  of  an  error  less  than  the  amount  OP 
is  then  equal  to  the  probability  of  the  occurrence  of  an  error 
greater  than  OP.  This  being  the  case,  OP  is  the  probable 
error  by  definition. 


CHAPTER  IV. 

COMPUTATION   OF   THE   PRECISION  MEASURES. 
/  DIRECT   OBSERVATIONS. 

59.  Observations  of  Equal  Weight.  Given  n  direct  obser- 
vations all  of  the  same  weight  on  a  single  quantity  M,  to 
find  the  Mean  and  Probable  Errors  and  Average  Deviation 
of  a  single  observation  and  of  the  Arithmetical  Mean. 

Let  the  observations  be  J/i,  M^,  .  .  .  Mn. 

u      "    arithmetical  mean  be  MQ. 

"      "    real  errors  be  x^    x2,  .  .  .    xn. 

"      "    residuals  be  vl5    v2»  •  •  •    vn- 

Denote  the  mean  and  probable  errors  and  average  devia- 
tion of  a  single  observation  by  /*,  r,  and  a.d.,  respectively, 
and  the  corresponding  quantities  for  the  arithmetical  mean 
by  fio,  r0,  and  A.D.  Then  by  definition, 


and 


/*  = 


If   MQ   represented    the    true  value   of    M,    the   residuals 
would  be  the  same  as  the  real  errors,  and  we  should  have 


/*  = 


and  if  n   is  large  this  formula  is  practically  exact.     Hut  when 
n    is  small    a    more  accurate   expression    is    necessary.     To 


48  METHOD  OF  LEAST  SQUARES. 

obtain  this  let   M0  -\-  x0   be  the  true  value  of   M.     Therefore 

Xi   =   Ml  —  (M0  -}-  x0)    =   vv  —  x0 
(-  *o)    =   vz  —  x0 


-j-  x0)  =  vn  —  x0 
Squaring,  adding,  and  dividing  by    n 

=  f  =      (2y2  -  2  x<£V  +  nx<? 


[-  ay2       for  by  (2)    2«  =  0 

The  value  of    x0    is    not  known  and  can  not  be   found 
exactly,  but  it  is  approximately  equal  to  the  mean  error  of 

Jf0,   that  is,  by   (48),   to   /to  =  JL. 

Substituting  this  value  in  the  above  we  have 

2    _ I      P" 

^  n      '    n 


*•  >  U*  . 

w 

by  (48) 
by  (42) 
and 

(n  -  l)^2  =   2u2 

(49) 

(50) 

(51) 
(52) 

fKit\ 

i/    2«?2 

IL     \/ 

V^_i 

t/      2v2 

U.       \/   - 

V  n(n-l) 

/M            flTl'i   i/ 

t/       2t?2 

/»«     —    .fi7,    i    \/ 

COMPUTATION  OF  THE  PRECISION  ME  AS  USES.       49 

60.    In    order   to   avoid   the   use   of  the    squares   of   the 
residuals  we  may  proceed  as  follows  :    From  (49) 


n 
On  the  average,  the  values  of  the  residuals  will  then  be 


/  n  -  1 
*  =  :  \  -^~  * 

I 

Vn    =     \ 

a          \ 


n  —  I 


n  —  1 


Adding  and  dividing  by    n,    neglecting  the  signs  of  the 
residuals, 


«H  0  \  ^  / 

n  \       n         n  \ 


n-l 
a.d. 


a.d.  =  (54) 

V/n(w--l) 

by  (48)  ^l.D.  =  -  (55) 

w  ^  n  —  1 

by  (41)  r=     >8453  Sty  (56) 

^n(n-l) 

.8453  i> 

and  r0  =  — inzi  (57) 

w-  Vw  —  1 

The  mean  errors  may  also  be  computed  from  the  above  by 
using  the  table  of  equivalents  in  paragraph  55,  but  this  is 
not  customary,  formulas  (50)  and  (51)  being  used  for  this 


50 


METHOD  OF  LEAST  SQUARES. 


purpose.  Results  derived  from  (50)  are  to  be  regarded  as 
more  accurate  than  those  obtained  from  (54),  the  latter 
being  a  second  approximation. 

61.  Example.  From  the  following  measurements  on  the 
length  of  a  base  line  find  the  most  probable  length  and  the 
values  of  the  various  precision  measures. 


M 

V 

y2 

455.35 

.02 

.0004 

.35 

.02 

4 

.20 

-  .13 

169 

.05 

-  .28 

784 

.75 

.42 

1764 

.40 

.07 

49 

.10 

-  .23 

529 

.30 

-  .03 

9 

.50 

.17 

289 

.30 

-  .03 

9 

3.30 

+  .70 

.3610 

455.330 

-  .70 

M* 

Sw  =  0 

Sw2 

By  (50)  and  (51) 


.3610 


r   =    .6745   p 
By  (54)  and  (55) 

1.40 
a.d.    = 

V/90 

r  ==  .8453  a.d. 


.20 
.13 

.15 
.13 


=  .063 


ra  =  .6745  u0  =  .042 


A'D-  =  IF    =  -047 


rn  =  =  .8453  A.D.  =  .040 


COMPUTATION  OF  THE  PEECISION  MEASURES.       51 

and  we  should  write  for  the   most   probable  length  of  the 
base  line 

J/o   =  455.330  ±  .042 

62.    Observations   of  Unequal  Weight.     Using  the  same 
notation  as  above,  with  slight  modifications,  we  will 

Let   MQ   represent  the  General  Mean. 
"    Pit  Pzi  •  •  •  Pn   be  the  weights. 
"     a.d.^  a.d.<i,  .  .  .  a.d.n   be  the  average  deviations. 
"     P-n   f*w  •  -  •  i*n  be  tne  mean  errors. 
"     r\->    r2i  •  -  -  rn  be  the  probable  errors. 
"     a.c?.,  /x,  r,  and  v  refer  to  observations  of  weight  unity. 

Then  by  (48) 

t  j. 

etc. 


If  the  "  Observation  Equations  "  are  formed  for  this  case 
they  will  be 

MI  —  Jf0  •=  v»     M2  —  3/o  =  -y2,  .  .  .  Mn  —  J/o  =  vn. 

And,  as  was  shown  in  paragraph  28,  if  these  equations  are 
each  multiplied  by  the  square  root  of  the  weight  of  the  corre- 
sponding observation,  they  will  all  be  reduced  to  equivalent 
equations  of  weight  unity.  On  performing  this  operation 
it  will  be  seen  that  the  residuals  of  the  new  equations 
become 


And  evidently  to  these  reduced  observations  the  formulas 
of  paragraph  59  apply. 


V 

52  METHOD   OF  LEAST  SQUARES. 

Therefore 

*pvl —          r  =  .6745  H          (58) 
n  —  I 


K   =    V  —  n  =  .6745  |i,         (59) 


=  N/ 


.         (60) 


Also,  by  a  method  similar  to  that  used  in  paragraph  60,  it 
may  be  shown  that 


a.d.  =  — =  r  =  .8453a.<?.       (61) 


a.eZ.fc  =    t  n  =  .8453  a.d.k     (62) 

n(w  —  1) 


.  J).  = 


—  I) 


Formulas  (61),  (62)  and  (63)  are  not  of  much  value  in 
practice  unless  all  the  weights  are  perfect  squares,  for  other- 
wise no  labor  is  saved  in  the  computations,  since  the  square 
root  of  each  weight  will  have  to  be  determined,  and  the  results 
obtained  can  not  be  considered  as  reliable  as  those  given  by 
formulas  (58),  (59)  and  (60). 


COMPUTATION  OF  THE  PRECISION  MEASURES.       53 


63.  Example.  Given  a  series  of  observations  on  M, 
difference  in  longitude  between  two  stations.  To  find 
/i,  /AO,  a.d.,  A.D.,  r,  riy  r0. 


the 


M 

p 

pM 

M 

v^/p 

V* 

pv2 

1°  4'  30" 

4 

120 

1.6 

3.2 

2.6 

10.4 

41 

1 

41 

9.4 

9.4 

88.4 

88.4 

43 

1 

43 

11.4 

11.4 

130.0 

130.0 

37 

9 

333 

5.4 

16.2 

29.1 

261.9 

48 

4 

192 

16.4 

32.8 

269.0 

1076.0 

34 

16 

544 

2.4 

9.6 

5.7 

91.2 

25 

9 

225 

6.6 

19.8 

43.6 

392.4 

46 

1 

46 

14.4 

14.4 

207.4 

207.4 

28 

25 

700 

3.6 

18.0 

13.0 

325.0 

24 

4 

96 

7.6 

15.2 

57.7 

230.8 

1°  4'  31".6 

74 

2340  1 

150.0 

2813.5 

Mo 

*P 

*p*4 

2  y^T* 

2/>y' 

M  = 


=   1°  4'  31".6 
=   17.7  a*  = 


r  =   .6745  a  =   11.9 


150 
a.d.   =  =    15.8 

V90 


V74 


r0   =    .6745  uo   =      1.4  .8453  A.D. 


=  2.1 

=    4.0 

1.8 
1.5 


M0  :       1°  4'  31".   6 


54        /  METHOD  OF  LEAST  SQUARES. 

FUNCTIONS   OF   OBSERVED   QUANTITIES. 

64.  Theorem.     Given  any  number  of  quantities  and  their 
Mean  and  Probable  Errors  and  Average  Deviations,  to  find 
the  Mean  and  Probable  Errors  and  Average  Deviation  of  auy 
function  of  the  quantities. 

Let  the  quantities  be        J/i,  M2,  .  .  .  Mq. 
"      "    mean  errors  be     /u.t,  ^  .  .  .  p.q. 
"      "    function  be          M=f(M1,M2,  .  ..  Mq). 
"      "    mean  error  of      M  be  E. 
"      "    probable  error  of  M  be  R. 
"      "    average  deviation  of  M  be  D. 

The  derivation  of  the  general  formula  will  be  simplified  if 
we  consider  first  a  few  special  forms  of  functions. 

65.  Case  I.     Suppose    Jff  =  _Mi  ±  J8fa. 

The  number  of  observations  from  which  Mv  and  M^  and 
hence  ^  and  ^  have  been  determined  is  not  necessarily 
known,  but  we  may  assume  that  for  each  quantity  it  is  any 
large  number  n,  and  that  the  real  errors  of  the  observations 
are 

for         J/i,     a;/,  a^",  «i'",  .  .  . 

u  Jf        x  '    X  "    X.,'" 

Then  the  real  errors  of  M,  computed  from  the  separate 
observations  on  J/i  and  M2  will  be 

y.    '   +    n.    >  ~    "    -|_     ~    "  rf    HI      \  III 

•*  \      ~^~    •*  "   5  \  '     *     9  \         ""*—        °          •    •    • 


n 


or  JE2  =   |t12+ti22  (64) 

since  in  the  most  probable  case  the  term    ^x^x^   will  dis- 
appear, as  there  most  likely  will  be  as  many  positive  as 


COMPUTATION  OF  TUB  PRECISION  MEASURES.       55 

negative   products  of  the   same  absolute  magnitude   of  the 
form   X-L  xz. 

By  successive  applications  of  the  above  the  same  principle 
may  be  extended  to  cover  the  algebraic  sum  of  any  number 
of  quantities.  So  that  if 

M  =   M,  ±  Mz  ±  .  .  .  Mq 
then       E*  =   m2  +   H^2  +  .  .  -  lV   =  2>2  («) 

Since  probable  errors  and  average  deviations  differ  from 
mean  errors  merely  by  a  constant  factor,  we  ishall  likewise 
have 

(66) 


.2      (67) 

66.  Example.     Given  the  telegraphic  longitude  results, 

h.  m.  sec.           sec. 

(A)  Cambridge  west  of  Greenwich,      4  44  80.99  ±  0/23 

(B)  Omaha            «      "Cambridge,       1  39  15.04  ±  0.06 

(C)  Springfield  east    "  Omaha,  25  8.69  ±  0.11 

Find   Z,,   the  longitude  of  Springfield,  and  its  probable  error. 

Z   =   A  4-  B  —  C  =   5*  58m  37'.34  ±  0'.26 
for  by  (66)          R   =   y/(.232  -f  .062  -f-  .II2)    =    .26 

67.  Case  IT.     Suppose     3£  =    a^f^ 

Using  the  same   notation   as  in    paragraph    64,    the    real 
errors  of  Jf  will  be 


n  :  f"1 

or  E    =     CT.Ji!  (68) 


56  METHOD   OF  LEAST  SQUARES. 

68.     Case  III.     Suppose   M  =  a\  Mv  ±  a2  M2  .  .  .  aq  Mq 
By  combining  (65)  and  (68) 


also  JJ2  =   SwV2  (69) 

and  J>2  =  S  a2  «.<?.' 

69.  Example.  The  length  of  a  bar  at  20°  Centigrade  is 
found  to  be  75".0041  ±  0".0037,  and  its  expansion  per 
centigrade  degree  0".0036  ±  0".0018.  What  is  the  length 
of  the  bar  at  56°  Fahrenheit  ? 

20°  C.    =    -  X  20  +  32    =    68°  F. 
5 

5  5 

The  expansion  for  1°  F.  will  be    -  X  -0036  ±  -  X  -0018 

a  9 

or  .0020  ±  .0010 

L  =  75.0041  -  12  X  .0020 
=  74.9801 


R  =  V-00372+  (12  X  .001)" 

=  .013 
L  =  74".980  ±  0".013 

70.     Case  IV.     Suppose   M  =  f  (Mi,   M*  •  .  .  Mq). 

Let  MI  =  «!-{-  »*i,  M2  =  «2  -j-  m2,  . . .  Mq  =  aq  -f-  wg, 
where  at,  a2,  .  .  .  ag  are  arbitrarily  assumed  quantities  very 
nearly  equal  to  M^  M^  .  .  .  Mq,  so  that  m^  m2,  .  .  .  mq  are 
so  small  that  their  second  and  higher  powers  may  be  neg- 
lected. Then  the  errors  of  Mlt  M2,  .  .  .  Mq  may  be  con- 
sidered to  be  in  ra^  m2j  .  .  .  mq,  and  hence  f^,  p.2,  ...  pq 


COMPUTATION  OF  THE  PRECISION  MEASURES.       57 

may   be    regarded    as   the   mean    errors    of   the    quantities 
»ij,  rw2,  .  .  .  mq.     We  now  have 

M  =  f  (<*i  +  mn  ««  +  wij,  .  .  .  «7  -|-  ™q) 

Expanding    this    expression    by   Taylor's    Theorem,    and 
denoting  f(ai,  a2,  .  .  .  a9)  by  J!/"',  we  have 

M  —   M' 

BM'  8M'  dM' 

-4-  m^  -= \-  mz  -= —  4-  .  .  .  ma  -= — •  (A) 

1    dat  2    ddt  q    daq 

-\-  negligible  terms    in    the   second    and 
higher  powers  of  m^  »?2,  .  .  .  mq. 

Then  the  mean  error  of   M  will  be  the  same  as  the  mean 
error  of  terms    (A),    and  by    (69)    this  is  given  by 


or  what  is  practically  the  same  thing,  by 


71.    Example  A.     Two  sides  of  a  right  triangle  are  meas- 
ured with  results 

a    =    49.53  ±  0.59 
b    =    50.38  ±  0.93  > 


58  METHOD  OF  LEAST  SQUARES. 

Find  the  length  of  the  hypotenuse    c    and  its  probable 
error.     In  this  example 


M  = 
dM 


y/  «2  -f    62 

By  (70)      J2-=    ^r  + 


T 

70.65  _70.65j 

E    =    .78 

and  c    —    70.65  ±  0.78 

.Example  B.  If  the  probable  error  of  x  is  r,  what  is 
the  probable  error  of  the  common  logarithm  of  x  ?  In  this 
case 

M  =  Iog10  MI    =   Iog10  a; 

9  l°gio  *  l°gio  e 

da;  a; 


Example  C.  If  the  weight  of  x  is  p,  what  is  the  weight 
p0  of  sin  x  ? 

Denoting  the  mean  errors  of  x  and  sin  x  by  /x.  and  JE', 
respectively,  we  have,  by  (45)  and  (70), 


COMPUTATION  OF  THE  PRECISION  MEASURES.       59 

Po  V? 

and  =    — 

p  E* 

P 
p0    =    -  —    =    p  sec8  x 

COS*  X 

72.  Equation  (70),  which  expresses  the  law  of  propaga- 
tion of  error  in  functions  of  observed  quantities,  is  one  of  the 
most  important  in  the  whole  theory  of  the  Method  of  Least 
Squares.     Upon  it  in  particular  is  based  the  discussion  of  th* 
"  Precision  of  Measurements."     This   subject  treats,  in  the 
first  place,  of  the   methods  of   finding  the    precision   of   a 
quantity  obtained  by  computation  from  a  series  of  measured 
quantities;  ?.nd,  in  the  second  place,  it  investigates  the  pre- 
cision with  which  the  component  measurements  of  a  series 
must  be  made  in  order  to  obtain  a  required  degree  of  pre- 
cision in  the  final  result.     The  following  simple  example  will 
illustrate  the  character  of  the  solutions  :  — 

73.  Example.     In  the   determination  of  a  current  bv  a 
tangent  galvanometer  we  have 

I  =    10  —  tan  <£ 
G 

where  I  is  the  current  in  amperes,  II  the  horizontal  com- 
ponent of  the  earth's  magnetic  force,  G  the  galvanometer 
constant,  and  <f>  the  angle  of  deflection.  Given  the  errors 
8i»  8j»  8s>  in  -">  @  an(l  tan  <£,  to  find  the  error  A  in  I. 

By  (70) 


100  100  //2  100  ZT2 

-—  V      (a) 


60  METHOD   OF  LEAST  SQUARES. 

100  //2 
Dividing  this  equation  by   1  2  =    -  tan2  </>   we  nave 


TAT    P-TJ  P° 
bJ  =  bJ  +  bJ 


That  is,  the  square  of  the  percentage  error  in  I  is  equal 
to  the  sum  of  the  squares  of  the  percentage  errors  in  H,  G, 
and  tan  <f>.  Hence  if 

If  is  determined  within  .4  per  cent. 
Q    u  n  «        .2     "       " 

tan  <f>   "  "  "        .1     "       " 

then  —  =   V  .16  -f-  .04  -j-  .01   =   .46  per  cent.      (c) 

Next,  suppose  the  value  of  I  is  required  to  within  .1  per 
cent.  To  find  the  necessary  accuracy  in  the  determinations 
of  H,  6?,  and  tan  <£  when  the  error  in  each  of  these  quan- 
tities is  to  have  the  same  influence  upon  the  total  error. 

From  (b)  we  shall  now  have 


A  =   -          =   .000577 


8j_   =   .00058^ 

82  =   .00058  G  (e) 

83  =    .00058  tan  <^> 

It  is  comparatively  easy  to  obtain  the  necessary  accuracy 
in  the  measurements  of  G  and  tan  <£,  but  difficult  in  the 
case  of  H. 

For  additional  work  of  this  kind  see  Holman's  "  Precision 
of  Measurements." 


COMPUTATION  OF  THE  PRECISION  MEASURES.       61 

74.  Combination  of  Functions  of  the  Same  Variables. 
It  is  to  be  noticed  that  equation  (70)  applies  only  when  M 
is  a  function  of  independent  quantities.  If  J/i,  M2,  .  .  .  Mq 
are  merely  different  functions  of  the  same  quantities  we 
must  proceed  as  follows  :  — 

Let  MI    =    <j>  (2X,  22,  .  .  .  zfc) 

M,    =    $  (2U  22,  .  .  .  «t) 
M    =  f  (  Jfte  Jff) 


If  any  single  observations  of  21?  z2,  .  .  .  2fc  are  subject  to 
errors  a^,  a;2,  .  .  .  ic^,  the  corresponding  errors  in  Jl/j  and 
J/2  will  be 


for   J/j,  Jfi    =    a1xl  -\-  «2«2  -(-...  a^xk  (a) 

"     J^,          JT2    =    a/Xi  -|-  a2'a;2  -f-  .  .  .  ak'xk  (b) 

Where  a^  az,  .  .  .  ak  are  the  differential  coefficients  of 
J/i,  and  «/,  «2'  .  .  .  at'  the  differential  coefficients  of  Jf2 
with  respect  to  2X,  22,  .  .  .  2^.  The  corresponding  error  in 
M  will  then  be 


X  =    AX:  +  A'X2  (c) 

Where    A   and    A   are  the  differential  coefficients  of   M 
with  respect  to    Mv   and    J^.     Substituting  in  (c)  from  (a) 
and  (b), 

X  =    (Aa-i.-\-  A'a\)  x^  -\-  (Aa^  -}-  A'a'z}  x2  .  .  . 

=     a  Xt    -\-    fix-i   -{-...    \Xk 


Then  if  the  number  of  observations  or  values  of   X  be 
denoted  by    «, 


.  .  .  XV.2  (d) 


62  METHOD  OF  LEAST  SQUARES. 

since    in   the   most   probable   case   the   product   terms   will 
cancel  out. 

Expanding  (d)   we  have 

E*  =    (Aa,  4-  .4V)2  Mi2  +  (Aa,  +  A'az')*  rf  +  .  .  . 

=  ^'(^v  -f  «2V  • .  •)  +  ^"KV  +  «2'V . . .) 
+  2yl^'(«i«iVi2  +  «2«2W+...)  (71) 

75.    Example.     As  a  very  simple  problem  take 

Jfi  =  22!,         3/2  =  82^         /*!  =  0.1, 
and  M  =  J/i  +  Jfs. 

Then      .4  =  1,         A'  =  1,         at  =  2,         at'  =  3. 
By  (71)  ^a  =  4  X  -01  +  9  X  -01  4-  2  X  2  X  3  X  -01  =  .25 
or  ^    =    0.5 

In  this  particular  example  the  result  may  be  found  directly 
from  (68)  by  substituting  at  first  in  M  the  values  of  M^ 
and  M.,. 

Thus  M  =   23!  4-  3zt   =   52t 

.E'  =    5/it   =    0.5 

If  M±  and  Jf2  had  been  independent  quantities,  by  (64) 
or  (69)  we  should  have  had 


E  =  V(2  X  0.1)2+  (3  X  O.I)2 
=   0.36 

INDIRECT    OBSERVATIONS. 

76.  The  determination  of  the  precision  measures  of  the 
unknown  quantities  in  case  the  observations  are  indirect 
involves  a  knowledge  of  the  weights  of  the  unknowns,  and 
consequently  the  method  of  computing  these  weights  must 


COMPUTATION  OF  THE  PRECISION  MEASURES.       63 

first  be  demonstrated.     It  will  be  assumed  at  present  that  all 
the  observations  are  of  weight  unity. 

FIRST  METHOD  OF  COMPUTING  THE  WEIGHTS. 
Let  the  observations  be 


in  which  M^  M^  .  .  .  Mn  denote  the  actual  observations, 
and  Zj,  z2,  ...  zq  the  most  probable  values  of  the  unknown 
quantities.  Let 

&!  —  J/i  =  mu     ki  —  M2  =  m^  .  .  .  kn  —  Mn  =  m^ 
Then  the  above  equations  give  rise  to  the 

OBSERVATION    EQUATIONS 


(A) 


By  the  rule,  paragraph  25,  we  now  form  the 

NORMAL    EQUATIONS 
2i  2  a2  -}-  22  2  ab  -j-  .  .  .  zv  2  ay  +  2  am    =    0 

zt  2  ab  _j_  22  2  *2   -f  ...  2^  2  i</  +  2  bm    =    0 

(B) 

21  2  a?  -f  22  2  bq  -j-  .  .  .  3?  2  (?2    +  2  ym    =    0 


64  METHOD    OF  LEAST  SQUARES. 

Multiply  the  first  of  (B)  by   Qly  the  second  of  (B)   by 
.  .  .  and  add  the  results.     Then 


«i  (£i2a2  -f  Q^ab  -f  .  .  . 
-f  z2  (Q^ab  +  &262   +  .  .  . 


4-  &S  am  -|-  ^22  6m  -f  ... 

=    0 


Let  §u  Qz,  ...  $g  be  determined  so  that  the  coefficient 
of  2a  in  (C)  shall  be  unity,  and  the  coefficients  of  za, 
38,  ...  zq  each  equal  to  zero.  That  is,  let 

&S  «2  +  #22  ab  +  .  .  .  Qql  aq  -  -   1    =    0 

=    0 


<2j2  aq  -\-  $22  bq  -}-•••   Qq^  q*  =0 

Then  (C)  becomes 

z^  _j_  Q^am  -|-  #226m  -f  .  .  .  QqSqm    =    0     (E) 

Equations  (D)  may  be  derived  from  the  normal  equations 
(B),  if  in  them  we  replace  zu  z2,  ...  2^  by  ^t,  $2,  .  .  .  ^g, 
2  «m  by  —  1,  and  2  6m,  2  cm,  ...  2  <?m  by  zero.  Hence 
the  solution  of  the  normal  equations  with  these  changes  will 
give  the  values  of  Qly  Q2,  ...  Qq. 

TO  SHOW  THAT   Qi  IS  THE  RECIPROCAL  OF  THE  WEIGHT  OF  XL 
Expanding  the  coefficients  of  (E)  we  have 


COMPUTATION  OF  THE  PRECISION  MEASURES.       65 


4-  «2™2  +  .  . 

4-  &(*!»zi  4-  *2w2  4-  •  •  •  *«™n 


=    0 
or  collecting  the  coefficients  of    m1}  wi2,  .  .  . 


=    0 
For  convenience  in  writing  we  will  let 

ax    =     Qlal   4-    ^2^!   4-    .  .  . 
03    =     ^Xa2   -j-    ^2i2    4    .  .  . 


(E') 


Zl    +    ajT^!    4-    02^2    4.    .  .  .    anWln     =      0  (G) 


Multiply  the  first  of    (F)    by    a1}    the  second  by    a2,  .  .  . 
and  add  the  results.     Then 


2oa    =    <2,2a2  4-  Q^ab  4-  ... 

=    1  by  first  of    (D)  (H) 

Multiply  the  first  of    (F)    by    blt    the  second  by    J2,  ... 
and  add  the  results.     Then 


66  METHOD    OF  LEAST  SQUARES. 


=    0  by  second  of    (D) 

Likewise         2  ca    =    0  (H') 


Multiply  the  first  of    (F)    by    at,    the  second  by    a2,  . 
and  add  the  results.     This  gives 

2a2     =      Qi2  aa  -j-   <>22  ba.  +    ...  $82  ?a 

=     &          ty  (H)  and  (H')  (I) 

Let    fi     be  the  mean  error  of  an  observation  of  weight  unity. 
Let    pzj  be  the  mean  error  of    zx. 
Let   pZl  be  the  weight  of    z^. 

The  mean  errors  of  m^  wz2,  .  .  .  mn  are  the  same  as  the 
mean  errors  of  J/i,  M^  .  .  .  Mn  and  each  is  accordingly 
equal  to  p.  Therefore  from  (G),  by  (69) 

Pv*    =    a!2/*2  +  02  V2  +  •  •  •  o«Va 

=     M22a2 

=    Ci/*'    by    (I)  (J) 

But  by  (48)    ^«    =    £- 

Pzi 

Comparing  this  with  ( J)   we  see  at  once  that 

Qi    =  (K) 

Pzi 

Therefore,  for  the  First  Method  of  computing  the  weights 
we  have  the  following  :  — 


COMPUTATION  OF  THE  PRECISION  MEASURES.       67 


77.  Rule  I.    In  the  normal  equation  for  z^   write  —  1 
for  the  absolute  term    2  am,    and  in    the  other  equations 
zero  for  each  of  the  absolute  terms    2  bm,  2  cm,  ...  2  qm. 
The  value  of  zl  found  from  these  equations,  is  the  recipro- 
cal of  the  weight  of  the  value  of  zx    obtained  by  the  solution 
of  the  normal  equations. 

To  Jind  the  weights  of  z2,  z3,  .  .  .  zq,  proceed  in  a  similar 
way,  forming  a  corresponding  set  of  equations  for  each 
unknown. 

SECOND  METHOD  OF  COMPUTING  THE  WEIGHTS. 

78.  Write  equations    (B)    of  paragraph  76  in  the  form 

zt  2  «2  +  z2  2  ab  -\-  .  .  .  zq  2  aq  -|-  2  am    =    A 
Zi  2  ab  -f-  z2  2  b*   -\-  .  .  .  zq  2  bq  -f-  2  bm    =    B 


bq  -f- 


qm    =    Q 


Then  in  the  solution  by  the  preceding  method,  equation 
(E)  becomes 


-f- 


(M) 


in  which,  as  was  proved  in  (K),  Ql  is  the  reciprocal  of  the 
weight  of  Zi.  Whatever  method  of  elimination  is  employed 
in  the  solution  of  the  normal  equations,  the  coefficient  of  A 
in  the  value  of  zx  must  necessarily  be  always  the  same. 
Hence  we  have 

79.  Rule  II.  Write  A,  B,  ...  Q  instead  of  zero  in  the 
second  members  of  the  normal  equations  and  carry  out 
their  solution  in  any  convenient  way.  Then  the  most 
probable  values  of  z^  z2,  .  .  .  z  are  yiven  by  those  terms 
in  the  results  which  are  independent  of  A,  Tt.  .  .  .  Q, 


68  METHOD  OF  LEAST  SQUARES. 

The  weight  of  z±  is  the  reciprocal  of  the  coefficient  of  A 
in  the  value  of  zx.  The  weight  of  z2  fa  the  reciprocal  of 
the  coefficient  of  JS  in  the  value  of  z2,  etc.,  etc. 

THIRD    METHOD    OF    COMPUTING   THE    WEIGHTS. 

80.  From  the  second,  third,  .  .  .  equations  of  (L)  find 
the  values  of  z2,  zs»  •  •  •  zq  in  terms  of  zv  and  substitute 
in  the  first  of  (L)  without  reduction.  Then  the  first  of 
(L)  becomes 

Rzi    =     T  +  A  +  terms  in  £,  (7,  ...  Q 

Where  T  is  the  sum  of  all  the  numerical  quantities  result- 
ing from  the  substitutions.  Dividing  through  by  Jl, 

T        A. 

zl    =   —  -f  —  -f  terms  in  B,  <7,  .  .  .   Q        (N) 
I\         Jf 

T 

in  which    —    is  the  most  probable  value  of    zt,    and,  as  was 
-B 

shown  in  deriving  the  second  method, 


R   =  A.  (O) 

From  this  follows  at  once 

81.  Rule  III.  Substitute  in  the  normal  equation  for  zt 
the  values  of  22,  zst  .  .  .  zq  in  terms  of  z±  as  found  from 
the  remaining  equations.  Then  before  freeing  of  fractions 
or  introducing  any  reduction  factor,  the  coefficient  of  2j 


COMPUTATION  OF  THE  PRECIS  ION  MEASURES.       69 

in  this  equation  is  the  weight  of  the  value  of  st  obtained 
in  the  solution. 

To  find  the  weights  of  22,  s3,  .  .  .  zq,  proceed  in  a  similar 
way  with  the  normal  equations  for  each  of  these  unknowns. 

For  the  solution  of  an  example  by  the  three  different 
methods  see  paragraph  84. 

THE  MEAN  ERROR  OF  AN  OBSERVATION. 

82.  The  next  step  will  be  to  derive  /x,  the  mean  error 
of  an  observation  of  weight  unity.  In  the  following  demon- 
stration the  equations  referred  to  by  letters  are  those  in 
paragraphs  76  to  81. 

Let  the  real  values  of    21}  22,  ...  zq    be 


and  substituting  in    (A)  we  have 

«i(2i  -h  «i)  +  &!(Z2  +  Je,)  -f  .  .  .  qi(zq  -f-  Xq}  -f-  mj  =   A! 

««(«i  +  ^i)  +  ^2(22  +  ^2)  +  •  •  •  ^O?  +  «ff)  +  ™2  =  A2 

(P) 

«n(2i  +  *!>  +  ftw(32  +  a;2)  -f  .  .  .  qn(zq  -f-  a?,)  4-  7nn  =  An 

where   Au  A2,  .  .  .  An  are  the  real  errors  of    J/i,  J/^,  .  .  .  J/J,, 
or  of   wij,  ra2,  .  .  .  mn. 

Multiply  the  first  of    (P)    by    au    the  second  by    «2,  ... 
and  add  the  results.     This  gives 

Zi  2  «2  -)-  z2  2  «A  -f-  ...  z7  2  «?  -f-  2  aw 
-j-  «i  2  a2  +  x2  2  «*  +  .  .  .  xq  2  ay  =    2  a  A 

But  by  the  first  of     (B)     the  first  line  in  this  equation  is 
equal  to  zero,  and  therefore 


70  METHOD    OF  LEAST  SQUARES. 

xl  2  a2  -j-  x«  2  ab  -f-  .  .  .  xq  2  aq  —  2  aA   =    0 
Also      xl  2  ab  +  x2  2  W  -f  .  .  .  xq  2  fy  -  2  £A   =    0 

.........  (Q) 

a?!  2  «?  +  x3  2  6?  -f-  .  .  .  xq  2  <?2  —  2  ?A   =    0 


These  being  of  the  same  form  as  the  normal  equations 
(B),  the  value  of  x^  resulting  from  their  solution  will  be 
of  the  same  form  as  that  of  z1  in  the  solution  of  those  equa- 
tions, with  only  the  substitution  of  —  A  for  m.  From 
(G)  we  shall  therefore  have 

X     —    a-i&     —    aA     —    ...   UjjAjj     =     0 


Multiply  the  first  of    (P)    by   vv,   the  second  of    (P)    by 
t>2,  .  .  .  and  add  the  results.     Then 


(z2  +  x^bv  +  .  .  .  (zq  -f- 


and  multiplying  the    first   of    (A)    by    a1}    the  second   by 
a2,  .  .  .  and  adding  the  results 

Say   =    Zj  2  «2  -f-  z2  2  ab  -j-  .  .  .  z7  2  a<?  -j-  S  aw 

=    0          by  the  first  of    (B). 
Also       2&y    =   0 

2?y    =   0 
Substituting  these  values  in  the  above,  we  find  that 

2  my    =    2  Aw  (S) 

Now   multiply  the   first   of    (A)    by    v^    the  second  by 
u    .  .  .  and  add  the  results.     This  shows  that 


COMPUTATION  OF  THE  PRECISION  MEASURES.       71 

2X  2  ay  -\-  z2  2  #y  -f-  •  •  •  z9  2  <?y  -|-  2  mu    =    5  u2 
and  as  above, 

2  ay    =    2£y...    =    2  ay    =    0 
Combining  this  result  with    (S)    we  have 

2  my    =    2y2    —    2  Ay  (T) 

Next  multiply   the    first  of    (A)    by    At,    the  second   by 
A8,  .  .  .  ,  add  the  results  and  compare  with    (T).     This  gives 

zl  2  aA  -(-  Z2  2  #A  -|-  ...  29  2  tfA  -f-  2  wiA 

=    2  Ay    =    2y2  (U) 

And  finally,  multiplying  the  first  of    (P)    by   Ax,    the  second 
by   A2,  ...  and  adding  the  results,  we  have 

«!  2  «A  -|-  22  2  &A  -j-  ...  2,  2  q\  +  2  mA 
+  «!  2  «A  +  x2  2  b\  -|-  .  .  .  2^2  ^A          =    2  A2 


Therefore  from    (U) 

2  y2  +  xl  2  aA  +  «,  2  6A  +  .  .  .  xq  2  ?A  =   2A2      (V) 


«    -  -         L   -  •  •  • 

^  w  ?i  w 

We  must  now  find  the  mean  values  of  the  terms   a^ 
JC2  2  iA,  ...  x9  2  </A.     Expanding    2  aA, 

2  aA  =  a^!  -f-  a2A2  -|-  .  .  .  anAn 
from  (R)  xl  =  a^  -(-  a2A2  -{-•••  ^^w 
Multiplying, 

Zj  2  aA    =    a^tAj2  -f  a-jOjA.,2  -f-  .  .  .  awanAn» 
-|-    terms  in    A,Aj,  AtAj,  .  .  . 


72  METHOD   OF  LEAST  SQUARES. 

In  the  most  probable  case  these  product  terms  vanish,  and 
substituting  for  A!*,  A22,  .  .  .  An2  the  mean  value  /*a,  we 
have 

xl  2  «A    =    /A2  2  aa 

=  M* 

Similarly  jc2  2  &A    =    /u,2 


From    (W)    then,    q   being  the  number  of  unknowns, 

,    =     2»2         qtS 
n  n 


I    T.,,a 

=    V-=^  •         (72) 

"  n  —  q 


*•    =    -7=    =    V;  ^  ('3) 


A.  

*   Z         • 


pz(n  -  q) 

83.    By  a  method  similar  to  that  used  in  paragraph  60,  we 
may  derive 


a.d.     =  —  (75) 

V  n(n  —  q) 


(76) 


\l  pzn(n  -  q) 


84.    Example.     In    illustration    of   the    above    processes 
take  the  example  in  paragraph  24,  where  we  found  for 


COMPUTATION  OF  THE  PRECISION  MEASURES.       73 
OBSERVATION    EQUATIONS 

Zl    —    22  —     1.7      =      Vi 

za  —  2.4    =    va  A. 

_  2l  +  z2  -L.  28  -   1.0    =    va 
22   —   Z3   —   3.0    =    v^ 
and  for 

NORMAL    EQUATIONS 

2  z1  —  2  22  -        23  —  0.7    =    0  (a) 

—  2  2t  4-  3  22  —  2.3    =    0  (b) 

-       2l  +  3  z3  -  0.4    =    0  (c) 


SOLUTION  FOR  THE  WEIGHTS  BY  THE  FIRST  METHOD. 

For  finding  the  weight  of   zt   the  above  normal  equations 
would  be  written 

2  zl  --  2sa  —      23  --   1    =    0               (a') 

-  2  zl  +  3  z*  =0               (b') 

«i                +  3  28  =0               (c') 

Solve  for   z 


3  X  (a') 

6  zi  -  6  z 

z  —  3  zs  —  3 

= 

0 

(c') 

Zl 

|           O     — 

= 

0 

5  Zi  —  6  z 

i                 -  3 

= 

0 

2  X  (V) 

_    4  2t    4-    6  2 

2 

= 

0 

2l 

-   3 

= 

0 

1 

2t      =      3 

Pzi      = 

¥ 

<*') 

For  finding  the  weight  of   22   the  equations  are 

2zt   --   2z,  -       2,  =0  (a") 

-  2  zt  +  3  2,  -1    =    0  (b") 

-  2i  +32,  =0  (c") 


74  METHOD  OF  LEAST  SQUARES. 

Solve  for  z, 

3  X  (a")  6  zl  --  6  22  —  3  zs    =    0 

(c")  _ji +  3  za    =    0 

5  zt  —   6  z2  =0 

6 

1  =    -  * 

5  3 

Substitute  in    (b")       z2    =  .-.      jt>Zi    =    -        (d") 

3  5 

For  finding  the  weight  of   zs   the  equations  are 

2z,   --  2z2  —      z^  =    0  (a"') 

-  2  Si  +  3  z2  =0  (b'") 

-  zx  +  3  z8  -   1    =    0  (c'") 

Solve  for   z8 

4 
from  (b'")  2  z2    =     -  zt 

'  O 

substitute  in  (a'")  2  Zj  —  3  z8  =0 

2  X  (c'")  -  2z,  +  6z3  --  2    =    0 

3  zs  -  2    =    0 

2  3 


SOLUTION  BY  THE  SECOND  METHOD. 

The  normal  equations  will  now  be  modified  so  as  to  appear 
in  the  following  form  :  — 

2zt   --   2  z2  -         z8  --   0.7    =    A  (a) 

-  2  zl  -4-  3  z2  -  2.3    =    B  (b) 

-       zl  -f  3  z8  -   0.4    =    C  (c) 


COMPUTATION  OF  THE  PRECISION  MEASURES.       75 
Solving, 

3  x  (a)          6  zl  -  6  z2  -  3  z8  -  2.1    =    3  A 

(c)    -      zl  _  -{-  3  g,  --  0.4    =    (7 

5  Zl  _  6  z2  --  2.5  =    3  ^4  -j-  C 

-  4  gt  -4-  6  z2  -  4.6  =    2  jg  _ 

2l  —  7.1  =    ZA  +  2J3+C     (8) 

Zi    =    7.1        and        ptl    =  (d) 

o 

Substituting  (S)  in   (b) 

3  z     =   16.5  _-6^         5^--2<7 


3 

22    =    5.5        and        jo^    =    —  (e) 

5 

Substituting  (S)  in  (c) 


zs    =    2.5        and        pZi    =  (f) 

« 

SOLUTION  BY  THE  THIRD  METHOD. 
The  normal  equations  are  now  taken  in  their  original  form. 

2  z,   --   2  z2  -         z3  --   0.7    =  0               (a) 

-  22!  +  3  z2                   -   2.3    =  0               (b) 

-  z,                 +  3  za  -  0.4    =  0                (c) 


76  METHOD   OF  LEAST  SQUARES. 

To  obtain  zt  and  its  weight  we  proceed  as  follows : 

zl  .4 

from  (c)  2,    =    —    +    — 

2  zl  2.3 

from  (b)  22    ==    — 1-    — 

Substitute  in  (a) 

4  4.6         2t          .4 

2  zv  —  -  Zi  _--_   —   ---—  0.7    =    0 
33  3  3 

zl         7.1 

Collecting  terms,  =    0 

o  o 

2l    =    7.1         and        pzi    =    -  (d) 

For  z2 
3  X  (a)  -j-  (c)          5  sjj  —  6  z2  —  2.5    =    0 

6 

zx    =    -  sa  +    0.5 
5 

12 

Substitute  in  (b) z*  —   1.0  -f  3  32  —  2.3    =    0 

5 

3 

Collecting  terms,       -  2a  —  3.3    =    0 
5 

3 

Zt    =    5.5         and        /»„    =    -  (e) 

For  2, 

3  X  (a)  +  2  X  (b)       2  2l  —  3  z3  -  6.7    =    0 

3  6.7 

2,       =      —  2.    +    

2    !          2 

O  ft    7 

Substitute  in  (c) 2,  — 1-  3  28  —  0.4    =    0 

—  2 


COMPUTATION  OF  THE  PRECISION  MEASURES.       77 

3  75 

Collecting  terms,  —  za  —  -        =0 

L  '— 

g 

zs    =    2.5         and         psa    =    •  (f) 

A 

It  is  evident  that  the  three  methods  give  identically  the 
same  results  and  that  the  work  is  about  the  same  in  each  case. 


COMPUTATION  OF  THE  PRECISION  MEASURES. 

Substituting   the  values   found   for    z^  z2  and  z3    in  the 
observations  equations  (A),  we  have 

7.1  _  5.5  -  1.7   =   Vt  =    -  .1         .01  =  v^ 

2.5  —  2.4  =   va  =  -j-  .1         .01  =  v22 

-  7.1  -f  5.5  +  2.5  -  1.0   =   vs  =    -  .1         .01  =  v,a 

5.5  -  2.5  —  3.0   =   v4  =         .0         .00  =  v42 


.03   = 
In  this  example     n  =   4,     q  =   3. 


By  (72)  p    =    y  =    .17 

By  (74)  r    =    .6745^  =    .12 


By  (73)       ^     =  =    .30  rzt     =    .20 

3 


§ 

9 

By  (75)  u.d.    =    —    =    .15 

<u 

r    =    .8453  a.d.    =    .13 


78  METHOD  OF  LEAST  SQUARES. 

85.  Observations  of  Unequal  Weights.  If  the  observations 
are  not  all  of  the  same  weight  the  formulas  and  operations 
are  merely  modified  in  the  usual  manner  and  equations  (72) 
and  (75)  take  the  more  general  form 


a.d.    =    — -  (78) 

V7  n(n  —  q) 


CONDITIONED   OBSERVATIONS. 

86.  Suppose  there  are  given  n  observations,  n'  condi- 
tions and  q  unknown  quantities.  Then  by  paragraph  33, 
the  method  of  solution  is  to  eliminate  n'  unknowns  between 
the  "  Observation "  and  "  Condition  "  equations,  leaving 
q  —  n'  independent  unknowns  in  the  "  Normal  "  equations. 
Consequently  formula  (77)  now  applies  to  this  case  and  it 
would  be  written 


=     y  (Jf) 

*  n  —  q  +  n 


also  p,   =    y  _  (80) 

v  p*(n  —  q  +  n  ') 

The  weights  of  the  q  —  n'  unknown  quantities  can  be 
found  by  any  one  of  the  three  methods  already  given  and 
then  the  mean  error  of  each  unknown  may  be  computed  by 
using  formula  (80).  If  the  mean  errors  of  the  n'  quantities 
that  were  first  eliminated  is  wanted,  their  weights  must  be 
determined  by  eliminating  a  different  set  of  n'  quantities 
from  the  original  observation  equations  and  solving  the  neces- 
sary sets  of  equations  for  these  weights.  The  first  method  of 


COMPUTATION  OF  THE  PRECISION  MEASURES.       79 

solution  for  the  weights  would  perhaps  be  best  here  as  the 
actual  values  of  the  unknowns  have  already  been  found. 
87.    Example.     Given  the 

OBSERVATION    EQUATIONS 

2l  -L.  22  —   3.0  =  0  weight  1 

2l              _  28  -f  1.5  =  0                  "4 

22  -  2.2  =  0                 "3        (A) 

Zl             -f  23  —  3.4  =  0                 "2 

and  the 

CONDITION    EQUATION 

2j}  _  22  -   0.5    =    0  (B) 

To  find  the  most  probable  values  of   zt,  z2,    and   zs,    and 
also  their  mean  errors. 

Eliminating   z8   between   (A)  and  (B)  there  remains 

Zl  _|-  22  —  3.0  =  0  weight  1 

Zl  -  z2  -f  1.0  =  0  "4 

z2  —  2.2  =  0  "3 

Zl  _|_  Z2  _  2.9  =  0  "2 

From  these  we  have  the 

NORMAL    EQUATIONS 

7  2l  -         z2  -     4.8    =    0 
-  zt  -f.  10  za  -  -   19.4    =    0 

Solving, 

Zl    =    0.98  pzi    =    6.9 

z2    =    2.04  pzt    =    9.9  (D) 

from  (B)          ZjJ    =    2.54 


80  METHOD   OF  LEAST  SQUARES. 

Now  eliminating   z2   between  (A)  and  (B),  we  find 

2t  _|_  28  _  3.5  =  0  weight  1 

2l  -  za  +  1.4  =<  0  «       4 

z3  -  2.7  =  0  «       3 

2l  _j_  2a  _  3.4  =  0  «       2 

From  these  we  derive  the  new  set  of 

NORMAL    EQUATIONS 

7  Zl  -         z8  -     4.3    =    0  p. 

-  zt  +  10  zs  —  24.4    =    0 

and  solving  for  z8 

zs    =    2.54  pZ3    =    9.9  (G) 

Substituting  the  values  of   Zj,  z2   and   z8   in  equations  (A), 
we  have  the  residuals 


V 

Vs 

<pv* 

.02 

.0004 

.0004 

.06 

.0036 

.0144 

.16 

.0256 

.0768 

.12 

-.0144 

-.0288 

.1204    = 
Since  in  this  example    n    =    4,     n'    =    1,     q    =    3, 


By  (79)  »   :  :V^      :   -25 


^    ==     -^  .09 

u,_    =     — ^—     =    .08 


p-zt    — =    -08 

\/9.9 


COMPUTATION  OF  THE  PRECISION  MEASURES.       81 

The  first  significant  figure  in  the  mean  errors  is  so  large 
that  it  is  not  worth  while  to  retain  the  second  place  as  usual. 

If  desired,  the  probable  errors  and  average  deviations  can 
now  be  computed  by  the  usual  formulas. 

88.  In  case  the  observations  are  made  directly  upon  the 
values  of  several  quantities  subject  to  certain  conditions,  we 
have  n  =  g,  and  equation  (79)  reduces  to 


v=^ 

from  which  the  mean  error  of  any  observation  may  at  once 
be  computed  from  its  weight. 

89.    Example.     Taking  the  example  in  paragraph  34  on 
the  measurement  of  the  angles  of  a  quadrilateral,  we  had  for 

OBSERVATION    EQUATIONS 

21    =    0  weight  3 

2 
2 

1 
for  the 

CONDITION    EQUATION 

Zl  -L.  z2  _|_  2j5  _j_  z4  4.  58    =    0  (B) 

and  for  the 

NORMAL    EQUATIONS 

4  zi  4-      z2  4-      z3  +  58    =    0 
*i  +  3  z2  4-      za  +  58    =    0  (C) 

2l  _j_      Z2  _j-  3  zs  +  58    =    0 

Solving  these  equations  for  the  values   of  the  unknown 
quantities  and  also  for  their  weights,  we  have 

Zi    =  8.3  pzi    —    3.5 

z2    =       -12.4  pzt    =    2.5 

zs    =       .    12.4  Pzs    =    2.5 


82  METHOD   OF  LEAST  SQUARES. 

and  forming  a  new  set  of  normals  containing   24,    we  find,  on 
solving  for  that  quantity, 

z4    =       -  24.9  =    1.75 


These  values  of  z,,  z2»  zv>  zt  are  a^s°  *n  this  case  the 
residuals  of  the  observations,  and  therefore  to  compute  the 
precision  measures  we  have 


V 

V* 

pv 

8.3 

68.9 

206.7 

12.4 

153.8 

307.6 

12.4 

153.8 

307.6 

24.9 

620.0 

620.0 

1441.9    = 
=    1 


By  (81)  M  =  =   38 

=   20  r2    =   13 


V/3.5 


=24  r^  =   rZ3   =   16 


=    29  r      =   20 


yl.75 

We  shall  accordingly  write  for  the  most  probable  values  of 
the  angles  of  the  quadrilateral 


A  = 

101° 

13' 

14" 

± 

13" 

B  = 

93 

49 

5 

± 

16 

ri   

87 

5 

27 

± 

16 

D  = 

77 

52 

15 

± 

20 

In  the  original  solution  in  paragraph  34  it  is  evident  that 
the  results  were  carried  out  to  a  greater  number  of  places  of 
significant  figures  than  the  character  of  the  observations 
warranted. 


CHAPTER  V. 

MISCELLANEOUS   THEOREMS. 

THE    DISTRIBUTION    OF    ERRORS. 

90.  Having  developed  the  processes  for  the  adjustment  of 
observations  according  to  the  Method  of  Least  Squares,  it 
will  now  be  interesting  to  show  how  closely  the  distribution 
of  errors  found  in  actual  practice  corresponds  to  the  theo- 
retical distribution  upon  which  our  methods  of  solution  are 
based. 

By  formula  (36),  the  probability  that  the  error  of  a  single 
observation  will  be  numerically  less  than  a  is 


p  =   T=      e~^dx  <82> 

VTT  Jo 

Let    t    =    hx,    .•.    dt    =    h  dx.     Also  when    x    =    0, 

t    =    0 ;    and  when    x    =    a,     t    =.    ha    =    p^L.      Sub- 

r 
stituting  in   (82), 


P   = 


p-'* 


=    ~  Cpre-'*dt  (83J 

VTT«Jo 

Values  of   P   for  values  of  the  argument   —   are  given  in 

T 

Table  I.  Also  for  any  series  of  observations  this  quantity 
P  will  represent  the  fraction  of  the  entire  number  which 
should  have  errors  less  than  the  amount  a.  Hence  if  P  is 
multiplied  by  the  whole  number  of  observations  the  result 
will  be  the  number  of  errors  which  should  be  less  than  the 
limit  a. 


84 


METHOD  OF  LEAST  SQUARES. 


91.  Example.  Forty  measurements  on  the  diameter  of 
Saturn's  ring  were  made  by  Bessel,  with  the  following 
results :  — 


M     v 

M     v 

M    v 

M    v 

38".91  -  .40 

39".35  +.04 

39".41  +.10 

39".02  -.29 

39  .32  +.01 

39  .25  —  .06 

39  .40  +-09 

39  .01  -  .30 

38  .93  -  .38 

39  .14  —.17 

39  .36  +.05 

38  .86  -  .45 

39  .31    .00 

39  .47  +.16 

39  .20  -  .11 

39  .51  +.20 

39  .17  —.14 

39  .29  -  .02 

39  .42  +.11 

39  .21  -  .10 

39  .04  —  .27 

39  .32  +.01 

39  .30  -  .01 

39  .17  -  .14 

39  .57  +.26 

39  .40  +-°9 

39  .41  +.10 

39  .60  +.29 

39  .46  +.15 

39  .33  +.02 

39  .43  +.12 

39  .54  +-23 

39  .30  -  .01 

39  .28  -  .03 

39  .43  +-12 

39  .45  +-14 

39  .03  -  .28 

39  .62  +.31 

39  .36  +.05 

39  .72  +.41 

From  these  the  most  probable  value  of  the  diameter  is 
found  to  be 

D    =    39".308  ±  0".022 

the  probable  error  of  a  single  observation  being    r  =  0".136. 
Compare  the  theoretical  and  actual  distribution  of  errors 


between 


u 

over 


0".00  and  0".05 

0  .05      "  0  .10 

0  .10      «  0  .20 

0  .20      "  0  .30 

0  .30      "  0  .40 
0  .40 


In  the  following  table  the  first  column  gives  the  successive 
values  of  the  limiting  error    a,    the  second  column  the  values 

a 
of     -,    and  the  third  column  the  corresponding  values  of   P. 


MISCELLANEOUS    THEOREMS. 


85 


The  fourth  column  contains  the  differences  between  the  suc- 
cessive values  of  f,  and  by  multiplying  each  of  these 
differences  by  40,  the  number  of  observations,  we  have  the 
quantities  in  column  five,  which  are  the  numbers  of  errors 
that  according  to  the  theory  should  fall  within  the  corre- 
sponding limits.  Column  six  shows  the  actual  number  of 
residuals  occurring  between  these  limits. 


a 

a 
r 

P 

d 

n 

n' 

0.00 

0.000 

0.000 

0.196 

8 

9 

0.05 
0.10 

0.368 
0.735 

0.196 
0.380 

0.184 
0.299 

7 
12 

6 
12 

0.20 

1.471 

0.679 

0.184 

7 

8 

0.30 

2.206 

0.863 

0.090 

4 

3 

0.40 

2.942 

0.953 

0.047 

2 

2 

00 

00 

1.000 

i 

This  is  a  close  agreement  considering  that  the  number  of 
observations  is  not  very  large.  Also  the  number  of  errors 
greater  than  the  probable  error  should  be  equal  to  the  num- 
ber less  than  it.  On  counting  the  residuals  we  find  twenty-one 
less  than  0".136  and  nineteen  greater. 


THE    REJECTION    OF   OBSERVATIONS. 

92.  After  a  series  of  measurements  have  been  made,  it  is 
frequently  found  that  one  or  two  of  the  observations  differ 
widely  from  the  others,  and  hence  it  becomes  a  matter  of 
great  importance  to  establish,  if  possible,  some  criterion  by 
which  we  may  determine  whether  such  discordant  observa- 
tions should  be  rejected  or  not.  We  are  not  concerned  here 


86  METHOD  OF  LEAST  SQUARES. 

with  the  question  of  the  detection  of  a  mistake  or  constant 
error,  which  a  consideration  of  the  circumstances  of  the 
observations  or  of  the  instruments  might  reveal,  but  it  is 
assumed  that  there  is  nothing  whatever  to  guide  us  except 
the  mere  fact  of  the  unusual  size  of  the  residuals  of  the 
observations  under  discussion.  To  reject  an  observation 
merely  because  it  differs  considerably  from  the  others  is 
entirely  unjustifiable,  while  to  retain  it  without  any  investi- 
gation is  a  neglect  of  the  evidence  furnished  by  the  observa- 
tions themselves. 

The  adoption  of  any  rigid  criterion  based  upon  the  magni- 
tude of  the  residuals  is  perhaps  more  satisfactory  from  a 
mathematical  standpoint  than  from  that  of  a  practical  observer, 
and  some  of  the  latter  are  of  the  opinion  that  no  observation 
should  be  rejected  entirely,  even  the  most  widely  discordant 
ones  being  given  a  certain  weight.  In  this  latter  case, 
however,  the  Theory  of  Probability  will  furnish  a  guide  as  to 
the  proper  weights  to  assign  to  the  different  observations. 

Of  the  various  criteria  that  have  been  proposed,  that 
developed  by  Pierce  (see  Chauvenet,  page  558)  is  perhaps 
the  most  complete.  The  derivation  and  application  of  this 
criterion  is,  however,  somewhat  long  and  complicated,  and 
for  all  ordinary  cases  the  following  simple  methods  will  give 
practically  as  good  results. 

93.  Criterion  for  the  Rejection  of  a  Single  Doubtful 
Observation.  It  was  shown  in  (83)  that  in  a  series  of  n 
observations  the  number  of  errors  numerically  less  than  a 
should  be  HP,  and  therefore  the  number  of  errors  greater 
than  a  should  be 

n  -  nP    =    n(\  —  P)  (84) 

If  the  value  of  the  expression  in  (84)  is  less  than  one-half, 
the  occurrence  of  an  error  of  magnitude  a  will  have  a 
greater  probability  against  it  than  for  it,  and  hence  the  obser- 
vation corresponding  may  be  rejected. 


MISCELLANEOUS   THEOREMS, 


87 


Accordingly  the  limit  of  rejection,    a,    of  a  single  doubtful 
observation  is  obtained  from  the  equation 


n  (1   -  P)    =    - 
V  '  2 


or 


P   = 


2  n  -  I 


(85) 


94.  JZxample.  Fifteen  observations  on  the  value  of  an 
angle  are  made.  Ought  any  of  the  observations  to  be 
rejected  ? 


M 

y 

V2 

v' 

v'2 

v" 

v"2 

2°     23'.90 

—    .30 

.090 

-  .41 

.168 

-  .33 

.109 

23.76 

-    .44 

.194 

-  .55 

.303 

-  .47 

.221 

25.21 

+  1.01 

1.020 

+  .90 

.810 

24.68 

+    .48 

.230 

+  .37 

.137 

+  .45 

.203 

23  .96 

-    .24 

.058 

-  .35 

.123 

-  .27 

.073 

24.26 

+    .06 

.004 

-  .05 

.003 

+  .03 

.001 

24.82 

+    .63 

.397 

+  .52 

.270 

+  .60 

.360 

24.07 

-    .13 

.017 

-.24 

.058 

-  .16 

.026 

23.98 

-    .22 

.048 

-  .33 

.109 

—  .25 

.063 

24.14 

-    .06 

.004 

-  .17 

.029 

-  .09 

.008 

24  .40 

+    .20 

.040 

+  .09 

.008 

+  .17 

.029 

24.38 

+    .18 

.032 

+  .07 

.005 

+  .15 

.023 

24.59 

+    .39 

.152 

-  .28 

.078 

+  .36 

.130 

24.10 

-    .10 

.010 

+  .21 

.044 

-  .13 

.017 

22.80 

-  1.40 

1.960 

2°     24  .20 

4.256 

2.145 

1.263 

j  METHOD   OF  LEAST  SQUARES. 

Using  all  the  observations  we  find 

3fft    =    2°    24 '.20         r    =    .6745  t  =    .37 


t/4-256    =    .J 
V      14 


OQ 

By  (85),  P    =    —    =    .967 

By  Table  I,          -    =    3.17      .-.      a    =    1.17 

As  the  residual  —  1.40  is  larger  than  a,  we  reject  the 
last  observation. 

From  the  remaining  observations  we  now  compute  a  new 
mean  value  and  a  new  set  of  residuals.  And  we  find 


o  145 
M'    —    2°  24'.31         r'    =    .6745  V/  — =    .27 


By    (85),  P   =    —    =    .964 

By  Table  I,          -    =    3.11      .-.      a    =    .84 


The  third  observation  may  accordingly  be  rejected. 
From  the  thirteen  observations  that  remain  we  find 


M"9    =    2°  24V23         r"    =    .6745  i/L263    =    .22 

v     12 


P    =         .    =    .962 
26 


-    =  3.08      .-.      a    =    .68 

r 


Therefore  no  more  observations  are  to  be  rejected. 


MISCELLANEOUS   THEOREMS.  89 

95.  The  Huge  Error.  In  cases  where  the  number  of 
observations  is  not  unusually  large,  a  simple  and  safe  criterion 
for  the  rejection  of  a  doubtful  observation  is  found  in  the  use 
of  the  "  Huge  Error." 

This  is  an  error  of  such  a  magnitude  that  999  out  of  every 
1000  errors  are  less  than  it  and  only  1  as  large  as  or  greater 
than  it. 

Therefore  the  probability  that  the  error  of  any  given 
observation  will  be  less  than  the  "  Huge  Error  "  is  .999,  and 
from  Table  I, 

when  P    =    .999,  —    =    4.9 

r 

a   =    Huge  Error   =    4.9  r 

=    3,3  jx  (86) 

=    4il  a.cl. 

Then  in  any  limited  series  of  observations,  if  an  error 
greater  than  the  huge  error  is  found,  we  should  reject  the 
observation  corresponding. 

See  also,  Holman,  page  30 ;    Wright,  page  131. 


CONSTANT    ERRORS. 

96.  Throughout  our  discussion  of  the  methods  of  adjusting 
observations  so  as  to  obtain  from  them  the  most  probable 
values  of  the  unknown  quantities,  all  constant  errors  are 
supposed  to  have  been  eliminated  before  the  Method  of 
Least  Squares  is  applied  in  deducing  the  results. 

If  this  is  not  done,  and  each  observation  is  subject  to  the 
same  constant  error,  the  final  result  will  be  affected  by  an 
equal  amount,  and  in  short,  the  Method  of  Least  Squares  is 
not  capable  of  removing  or  reducing  the  effect  of  errors  of 
this  kind.  All  that  is  accomplished  by  the  use  of  the  method 
is  to  reduce  to  a  minimum  the  effect  of  the  Accidental  Errors. 


90  METHOD  OF  LEAST  SQUARES. 

Hence  it  will  be  seen  that  although  by  increasing  the 
number  of  observations  of  a  given  kind  we  may  increase  the 
precision,  that  is,  reduce  the  probable  error,  of  our  final 
result  as  much  as  we  choose,  yet  we  do  not  in  this  way 
necessarily  increase  the  accuracy  of  the  determination. 

But  if  the  unknowns  can  be  determined  in  several  ways,  or 
under  a  variety  of  different  circumstances,  with  various 
instruments,  or  by  different  observers,  then  it  is  most  proba- 
ble that  the  constant  errors  of  the  different  sets  of  measure- 
ments will  be  grouped  about  the  true  values  of  the  unknowns 
according  to  the  exponential  law  of  error.  Accordingly  a 
combination  of  such  observations  will  enable  us  to  increase 
not  only  the  precision,  but  also  the  accuracy  of  the  final 
result,  the  constant  errors  of  the  different  sets  tending  to 
cancel  each  other  in  the  same  way  that  the  accidental  errors 
of  a  single  set  do. 

It  is  for  this  reason  that  determinations  of  a  quantity  from 
observations  made  in  a  variety  of  ways  are  more  valuable 
than  those  obtained  merely  from  different  sets  of  measure- 
ments of  the  same  kind. 

97.  The  probability  of  the  existence  of  a  constant  error 
may  often  be  expressed  in  the  following  manner. 

Example.  A  standard  100  ohm  coil  is  compared  with 
a  Wheatstone's  bridge  and  the  mean  result  found  to  be 
100.90  ±  0.20.  To  find  the  probability  that  there  is  an  error 
in  the  bridge  between  -|-  0.30  and  -(-  1.50  ohms. 

Suppose  the  result  100.90  ±  0.20  is  treated  as  a  single 
observation,  and  we  find  by  an  application  of  (83)  the  prob- 
ability that  the  error  of  this  observation  is  numerically  less 
than  0.60  ohms. 


.f\ 

Here  -    =    —    =    3.00      .-.       P    =    .957 

r  .20 

Hence,  as  far  as  is  shown  by  the  observations,  the  proba- 
bility that    100.90  ohms    is  within    0.60  ohms    of   the    true 


MISCELLANEOUS    THEOREMS.  91 

value  is  .957.  But  since  it  is  known  that  the  true  resistance 
is  100  ohms,  it  follows  that  there  is  the  same  probability 
that  there  is  a  constant  error  in  the  bridge  between  -|-  0.30 
and  -|-  1-50  ohms. 

98.  Combination  of  Determinations  having  Different  Con- 
stant Errors.  In  case  two  or  more  determinations  of  a 
quantity,  together  with  their  probable  errors,  are  obtained, 
the  method  of  combining  them  so  as  to  secure  the  best  final 
result  was  considered  in  paragraph  56,  and  in  Example  C, 
paragraph  57.  But  it  was  there  assumed  that  all  the  results 
were  subject  to  the  same  constant  errors,  while  if  this  is  not 
true  the  probable  errors  of  the  separate  determinations  bear 
no  relation  to  their  weights,  and  accordingly  in  such  cases 
another  process  must  be  adopted. 

To  determine  whether  the  different  measurements  may 
fairly  be  considered  to  have  the  same  constant  errors  we  may 
proceed  as  follows  :  — 

Let  the  determinations  of  the  quantity   M  be 

-fl/i  ±  ^  (a) 

MI  ±  rz  (b) 

and  let  the  difference  between  these  results  be 

d    =    J/i  —  Hfa  (c) 

Then  the  probable  error  of    d    is  by  (66), 


72    =    vV  +  r,«  (d) 

If  d  is  of  such  a  magnitude  that  an  accidental  error  as 
great  as  it  may  reasonably  be  expected,  we  may  assume  that 
the  constant  errors  of  (a)  and  (b)  are  the  same,  and  pro- 
ceed as  in  Example  C,  paragraph  57. 

But  if  the  probability  of  making  two  determinations  which 
differ  by  the  amount  d  is  very  small,  we  had  best  consider 
J/t  and  MI  to  have  the  same  weight,  provided  there  are  no 


92  METHOD  OF  LEAST  SQUABES. 

special  reasons  for  regarding  one  better  than  the  other.  The 
final  value  ot  M  will  then  be  the  arithmetical  mean  of  MI 
and  JH/2,  and  its  probable  error  will  be  found  by  (53). 

99.    Example  A.     An  angle  is  measured  by  a  theodolite 
and  by  a  transit  with  results 

By  Theodolite,     24°    13'  36".0   ±     3".l 
By  Transit,          24°    13'   24"      ±    14" 

What  is  the  most  probable  value  of  the  angle  and  its  prob- 
able error? 

Referring  to  the  preceding  paragraph, 

^    =    3.1,         r2    =    14,         d    =    12, 
and  the  probable  error  of    d  is 


R    =   V3.12  +  142    =    14 

Then  from  Table  I  the  probability  that  the  accidental 
error  of  a  determination  will  be  at  least  as  large  as  12  is 
found  from 

*-    =     1?     =    .86        ,.         1   -  P    =    .57 
r  14 

That  is,  there  is  more  than  an  even  chance  that  two  such 
determinations  of  the  angle  will  differ  by  as  much  as  12. 

Hence  it  is  fair  to  assume  that  the  two  determinations  are 
not  affected  by  constant  errors  of  different  magnitudes,  and 
they  would  be  combined  as  in  Example  C,  paragraph  57. 

Example  J3.  Suppose  the  zenith  distance,  M,  of  a  star, 
observed  at  two  different  culminations,  is  found  to  be 

J/i    =    14°    53'    12".10    ±    0".30 
M2    =    14°    53'   14".30   ±   0".50 

What  is  the  best  final  value  ? 


Here          d   =    2.2,         72    =    ^  .09  -f  .25    =    .58 

and  for        -    =    —    =    3.8,         1   -   P    =    .01 
r  .58 


MISCELLANEOUS   THEOREMS.  93 

Therefore  the  chance  that  the  difference  in  the  two  deter- 
minations, due  to  accidental  errors,  will  be  as  large  as  2.2  is 
only  one  in  a  hundred.  It  is  to  be  concluded  then  that  the 
constant  errors  of  observations  at  the  two  culminations  differ 
by  about  2.2,  and  as  there  is  nothing  to  show  that  one  meas- 
urement is  more  accurate  than  the  other  we  will  give  them 
both  the  same  weight  and  take  the  mean.  Then  the  best 
value  for  the  zenith  distance  is 

M0    =    14°   53'    13".20   ±   0".74 

For  M.    =    14-    53-   +     13''10  +  14"3° 

2i 

=    14°    53'   13".20 


2.42 
By  (53)        r0    =    .6745  V^-^;    =    -74 

For  a  more  extended  treatment  of  this  subject  see  Johnson, 
"  The  Theory  of  Errors  and  Method  of  Least  Squares," 
chap.  vii. 

THE    WEIGHTING   OF    OBSERVATIONS. 

100.  In  case  the  relative  worth  of  observations  is  not 
settled  by  methods  already  discussed,  the  proper  weight  to 
assign  to  each  quantity  in  the  final  adjustment  can  only  be 
determined  from  a  full  knowledge  of  all  the  circumstances  of 
the  measurements.  Even  then  considerable  experience  in 
the  particular  work  in  hand  is  required  before  the  best  values 
for  these  weights  can  be  assigned.  The  weight  given  to  a 
quantity  should  never  be  considered  final,  but  always  subject 
to  revision  whenever  new  information  with  regard  to  the 
quantity  is  obtained.  Thus  an  observation  which  at  first  is 
supposed  to  deserve  a  high  degree  of  confidence  is  often 
found  on  later  investigation  to  possess  very  little  value,  and 
vice  versa. 

See  also  Wriyht,  page  118. 


94  METHOD  OF  LEAST  SQUARES. 

OTHER    LAWS   OF    ERROR. 

101.  Although  in  the  great  majority  of  cases  the  distribu- 
tion of  errors  follows  the  exponential  law  thus  far  considered, 
there  are  a  few  special  cases  in  which  some  of  the  suppositions 
made  in  deriving  that  law  do  not  hold,  arid  hence  for  the 
adjustment  of  such  observations  the  corresponding  special 
laws  of  error  must  be  determined. 

For  instance,  in  applying  the  exponential  law  we  assume  a 
large  number  of  observations,  that  each  observation  is  sub- 
ject to  the  same  law  of  error,  that  small  errors  are  more 
likely  to  occur  than  large  ones,  and  that  positive  and  negative 
errors  are  equally  probable.  Now  it  is  easy  to  conceive  of 
cases  where  only  positive  errors  can  occur,  or  where  the 
probability  of  the  occurrence  of  a  small  error  may  not  be 
greater  than  that  of  a  larger  one,  etc.  If  we  can  determine 
the  different  sources  of  error  in  any  case  and  the  relative 
effect  of  each  upon  the  quantity  sought,  we  shall  arrive  at 
the  law  of  error  for  that  particular  set  of  observations.  The 
case  of  most  common  occurrence  is  the  following. 

102.  Suppose  all  errors  between  the  limits    a   and   —  a 
are  equally  probable,  and  that  there  are  no  errors  beyond 
these  limits.      Then  if    y  =   <f>(x)    is  the  equation  of  the 
Curve  of  Error,   and  its  area   is    represented   as   in    para- 
graph 18,  we  have 


<f>(x)  dx    =    1  (a) 

or  2  <t>(x)  I     dx    =    1  (b) 


since  by  the  supposition  made    <f>(x)    must  be  a  constant. 
Integrating  and  solving  for   </>(«),  we  have 


y   = 


*  ct 


MISCELLANEOUS   THEOREMS. 


95 


To  find  the  Mean  Error  we  have  by  definition  as  in  para- 
graph 52 

p*    =      Ca  x2  <£(a;)  dx 

«/  -a 


_    i   Ca 

a  Jo 


x*  dx 


a 

7? 


The  Probable  Error  is  derived  from  the  equation 

Cr  1 

<£(a:)  dx    =    — 

J -r  2 

1     CT  j  1 

_    I     dx    =    — 

a  c/o  9 


Finally,  for  the  Average  Deviation  we  have 

/"*flt 

a.d.    =          x  <£  (x)  dx 

J  -a 

=    _   I     x  dx 

a  Jo 

7 

And  the  Curve  of  Error  has  the  form 


(88) 


(89) 


(90) 


96  METHOD  OF  LEAST  SQUARES. 

That  the  average  deviation  and  probable  error  are  in  this 
case  equal  to  one  half  of  a  may  also  be  seen  from  the  defini- 
tion of  these  quantities. 

Example.  In  taking  a  logarithm  from  a  four  place  table, 
what  is  the  probable  error  of  the  mantissa  ? 

In  this  case  the  maximum  error  is  .00005,  and  all 
errors  between  —  .00005  and  .00005  are  equally  probable. 
Therefore 

r    =    .000025 

103.  The  only  other  special  case  of  common  occurrence  is 
that  in  which  the  error  of  a  quantity  is  due  to  two  sources, 
each  of  which  can  with  the  same  probability  assume  all 
values  between  a  and  —  a.  Here  it  may  be  shown  that 
the  curve  of  error  consists  of  two  straight  lines  whose  equa- 
tions are 

2a  —  x  2a  -\-  oc 

V   =    -£;-      and      „    =    —  _     (91) 


Also      (i1   =    -    a?,  r   =    (2  -  <fz)  a      (92) 

3 

For  a  more  extended  discussion  of  special  laws  of  error,  see 
"Wright,  paragraphs  31  to  39.  See  also  Example  151. 

104.  Contradictory  Observations.  Suppose  three  observa- 
tions of  a  quantity  give  results  55,  56,  91.  It  is  obvious  that 
to  take  the  arithmetical  mean  as  the  most  probable  value  in 
this  case  would  be  contrary  to  the  evidence  furnished  by  the 
measurements,  while  in  so  small  a  number  of  observations  it 
would  not  be  allowable  to  reject  the  value  91  entirely.  With 
such  a  series  of  observations  no  satisfactory  solution  can  be 
obtained,  but  probably  the  best  thing  to  do  would  be  to  take 
the  value  which  has  as  many  observations  less  than  it  as  it 
has  greater,  or  56. 


CHAPTER  VI. 

GAUSS'S   METHOD   OF   SUBSTITUTION. 

105.  The  most  laborious  part  of  the  application  of  the 
Method  of  Least  Squares  to  the  adjustment  of  observations 
consists  in  the  formation  and  solution  of  the  "Normal  Equa- 
tions," and  this  labor  increases  enormously  with  increase  in 
the  number  of  observations,  of  unknowns  and  of  conditions. 
It  is  not  at  all  unusual  to  find  that  the  adjustment  of  a  single 
set  of  observations  takes  several  weeks,  even  with  all  the  aid 
that  can  be  obtained  from  tables  of  logarithms,  of  squares,  of 
products,  and  of  reciprocals,  and  also  from  the  use  of  calcu- 
lating machines.     In  general  the  computations  can  be  per- 
formed more  rapidly  and  with  less  fatigue  by  using  a  machine 
or  a  table  of  products  than  by  using  logarithms,  but  a  combi- 
nation of  methods  is  often  desirable. 

106.  In  any  case  however  it  is  of  the  utmost  importance 
that  the  formation  and  solution   of   the   normal   equations 
should  be  effected  in  a  systematic  way,  and  that  as  far  as 
possible  checks  on  the  numerical  work  be  carried  along  in 
the  computations.     By  a  slight  amount  of  additional  work 
checks  upon  the  results  at  successive  stages  of  the  solution 
may  be  obtained  by  means  indicated  in  the  following  dem- 
onstrations, while  the  most  satisfactory  form  for  the  solution 
of  the  normal  equations  is  given  by  the  "Method  of  Substitu- 
tion" proposed  by  Gauss. 

This  method  will  now  be  explained;  and  it  is  to  be  observed 
that  it  is  customary  in  this  subject  to  enclose  a  quantity  in 
brackets  when  the  sum  of  a  number  of  quantities  of  the  same 
kind  is  to  be  denoted.  Thus  [«£>]  means  the  same  as  2«6. 

For  the  sake  of  simplicity  in  demonstration  it  will  be 
assumed  that  the  observations  are  all  reduced  to  weight  unity. 


98  METHOD  OF  LEAST  SQUARES. 

107=    Checks  on  the  Formation  of  the  Normal  Equations. 
If,  as  in  paragraph  76,  we  take  for 


OBSERVATION    EQUATIONS 


-f-  *2«2  +  •  •  •  Q&q  +  m2    =    «2 

(A) 


we  shall  have  for 

NORMAL   EQUATIONS 


?  +  [aw]    =    0 
[aft]  %  +  [ftft]  «,+  ...  [fty]  z9  +  [ftm]    =    0 

.........  (B) 


i  +  [ft?]  «t  +  •  .  •  [??]  zg  +  [?m]    =    0 
Let 

«i  +  *i  +  •  •  .  q\  +  mi  =   5i 

«8    +    *2    +    •  '   •     2*    +    ^2      =       *2 

.........  (C) 

an  -f-  ft»  -f-  .  .  .  qn  -j-  mn    =    sn 
...      [a]  +  [ft]  +  .  .  .  [?]  4.  [m]    =    [s] 


Multiplying  the  first  of   (C)   by  7W1?   the  second  by  w2,  .  .  . 
and  adding,  there  results 


[am]  +  [6w]  -j-  .  .  .  [gra]  +  [wiwi]  =  [«m]      (93) 

Next  multiplying  each   of   equations    (C^   by  its    a   and 
adding,  and  then  each  by  its   b   and  adding,  etc.,  we  have 


GAUSS'S  METHOD  OF  SUBSTITUTION.  99 


[cm]  +  [«6]  +  .  .  .  [ag]  +  [«»&]  =   [«*] 
[aft]  +  [&&]  +  .  .  .  [6g]  +  [&w]  =   [&«] 

.........  (94) 

[ag]  +  [6g]  +  .  .  .  [gg]  -}-  [gm]   =   [g«] 

Equation  (93)  will  be  satisfied  if  the  absolute  terms  in 
the  normal  equations  are  correct,  and  equations  (94)  when 
the  coefficients  of  the  unknown  quantities  are  correct.  These 
check  the  formation  of  the  normal  equations. 

108.  The  Reduced  Normal  Equations  and  the  Elimination 
Equations. 

The  value  of  zl  in  terms  of  the  remaining  unknowns, 
derived  from  the  first  of  equations  (B),  is 

[oft]  [oc]  [am] 

'   [aa]    2  "   [aa]  Z*  ~          '   [aa] 

Substituting  this  in  the  remaining  n  —  1  equations,  they 
become 


-  [ao]  +  .    .    [cm]  -  [«o]  =  0 

L          J  ' 


-f 

L      J   [aa] 

And  letting 


(F) 


100  METHOD   OF  LEAST  SQUARES. 

the  above  equations  take  the  following  form,  which,  being 
the  same  as  that  of  the  original  normal  equations,  they  are 
called  the 

FIRST  REDUCED  NORMAL  EQUATIONS, 

[ftfl,  1]  Z2  +  [ftc,  1]  Z3  +  .  .  .  [fy,  1]  zq  4-  [ftm,  1]  =    0 

[ftc,  1]  z2  +  [cc,  1]  2,  +  .  . .  [eg-,  1]  zg  4-  [cm,  1]  =   0 

(G) 


\bq,  1]  22  4~  [C3S  1]  zs  4~  •  •  •  [?!?>  1]  sg  4~  [?m»  1]   =    0 

An  inspection  of  equations  (F)  will  render  it  easy  to  form 
a  rule  for  writing  out  any  one  of  them. 

Now  by  means  of  the  first  of  equations  (G),  eliminating 
z2  from  each  of  the  others  in  the  same  way  that  z^  was 
eliminated  from  the  normal  equations,  there  results  the 

SECOND  REDUCED  NORMAL  EQUATIONS 

[cc,  2]  z3  4~  •  •  •  [c<7,  2]  zq  4-  [cm,  2]    =    0 

(H) 

[eg,  2]  z3  4~  •  •  •  [?<?>  2]  zq  4~  [$^>  2]   =    0 
In  which 

[ftc,  1]  [£c,  1] 


[cc,2]     = 


,  1] 

(I) 


Continuing  this  process  we  shall  finally  arrive  at  the  single 
equation 

-1]  =  o  (J) 


from  which  the  value  of   zq   is  determined. 


GAUSS'S  METHOD  OF  SUBSTITUTION.  101 

The  value  of  zq_±  will  then  be  obtained  by  substituting  the 
numerical  value  of  zq  in  the  first  of  the  preceding  set  of 
equations,  and  so  on,  until  finally  2t  is  obtained  from  the 
first  of  the  original  normal  equations.  The  equations  from 
which  the  unknowns  are  actually  determined  are  then  the 
following,  called  the 

ELIMINATION   EQUATIONS. 


-f-  .  .  .     [a#]  zq  -f-  [aw]  =  0 
]*ff  +  [6m,  1]  =  0 

(95) 

«  4-  [«™>  q-i]  =  o 

It  may  be  seen  from  the  rule  in  paragraph  81  that 
»  <1  —  1]  i8  ^e  weight  of  2g,  and  the  weight  of  any 
unknown  might  be  found  at  the  same  time  as  its  value  by 
making  it  the  last  in  the  order  of  elimination,  but  except  in 
special  cases  the  weights  had  best  be  obtained  by  the  general 
process  of  paragraph  115. 

109.  Check  on  the  Solution  of  the  Normal  Equations. 
Multiplying  the  first  of  the  Observation  Equations  (A)  by 
mt,  the  second  by  m2,  .  .  .  and  adding  the  results,  we  have 

[my]    =    [am]  zt  -j-  [6m]  22  -f~  •  •  •  \.<lm~\  zq  ~\~  [W4>"] 

But  in  equation  (T),  paragraph  82,  it  was  shown  that 
[my]  =  [vy].  Therefore 

[yy]    =    [am]  zl  -j-  [6m]  22  -(-...  \_qni]  zq  -f-  [mm] 

Substituting  in  this  the  value  of  zl  from  the  first  of  (95), 
we  get  the  result 

[ww]    =    [6m,  1]  za  -|-  [c/»,  1]  z3  -f  .  .  . 
in  which 


102  METHOD   OF  LEAST  SQUARES. 

n          i-i  n      n  Ca^l   [«W*1 

[bm,  1]    =     [im]   —    L    _J  L      J 

[aa] 


r          in  r        -> 

[mm,  1]    =  [mml  — 


(K) 

[am]  [am] 

[(/(/] 


being  similar  in  form  to  equations  (F). 

Next  eliminating   za    in  a  like  manner,  we  get 

[uw]    =    [cm,  2]  a,  +  .  .  .  [?m,  2]  z?  -J-  [mm,  2] 
and  continuing  this  process  it  finally  appears  that 

[yv]    =    [mm,  q]  [96] 

110.  Arrangement  of  the  Computations.  In  computing 
the  coefficients  that  appear  in  the  "Auxiliary"  or  "Reduced 
Normal  Equations"  it  is  most  convenient  to  arrange  the  work 
in  tabular  form.  The  arrangement  of  the  solution  will  be 
illustrated  for  an  example  containing  four  unknowns  but  it 
will  be  evident  that  the  process  can  be  extended  to  cover  any 
case. 

Let  the  Observation  and  Normal  Equations  be  represented 
by  equations  (A)  and  (B),  there  being  only  four  unknowns 
z\i  22>  zs>  Z4»  an(i  arrange  a  table  as  on  the  next  page.  In  this 
scheme  of  solution  the  upper  lines  of  the  rows  in  the  first 
compartment  contain  all  the  quantities  that  appear  in  the 
Normal  Equations,  together  with  [mm]  and  the  quantities 
in  the  column  headed  *  which  are  used  in  checking  the 
results  in  accordance  with  equations  (93)  and  (94).  The 
other  compartments  contain  the  corresponding  quantities 
for  the  Reduced  Normal  Equations,  and  the  first  line  in 
each  compartment  gives  the  coefficients  in  the  Elimination 
Equations. 


GAUSS'S  METHOD  OF  SUBSTITUTION.  103 

SCHEME  A, — SOLUTION  OF  THE  NORMAL  EQUATIONS. 


a                         b                          e                         d                        m                          I 

[«]            0*] 

log  [on]             log  [aA] 

[ae] 
log  [ae] 

[arf] 
log  [arf] 

[am] 
log  [am] 

log  [at] 

\0gAb           log^A[aA] 

log  ^6  [acl 

[4rf] 

[4m] 
At,  [am] 
log  Ab  [am] 

[4,]' 
log  Ab  [a»] 

log  At 

[ec] 
log  Ac  [ac] 

[erf] 

[em] 
Ae  [am] 
log  Ac  [am] 

W 

AM 

log  Ae  [as] 

log  Jrf[arf] 

[rfm] 
At  [am] 
log  AJ  [am] 

[rf5] 

4«M 

log  ^rf  [aj] 

[mm] 
Am  [am] 
log  Am  [am] 

log  J.,  [at] 

[44,  1] 
log[«,  1] 

[4*.  1] 
log[6e.l] 

[Arf,l] 
Iog[4rf,  1] 

[fa,  I] 

log  [4m,  1] 

log  [4*.  1] 

log  B. 
log  5. 

[ee,  1] 
log's,  [be,  1] 

[erf,  1] 

lof/t^l] 

[c»,  1] 
log  \  [4s,  1] 

B*  [W.  1] 

[rfm,  1] 
^,[4m,  1] 
log  Z?,f  [4m,  1  ] 

[rfs,  1] 
^  [4*.  1] 

[mm,  1] 

bf^Jn 

logC. 

0-  2] 
log  [ec,  2] 

[erf,  2] 
log  [erf.  2] 

[cm,  2] 
log  [cm,  2] 

[«•  2] 
log  [ci,  2] 

[rfrf,2] 
C'_[crf,2] 
log  C,  [erf,  2] 

[rfm,  2] 
Cd  [em,  2] 
logC,,[em,2] 

[rf»,  2] 
Cd  [e«,  2] 
log  C,  [e,,  2] 

[mm,  2] 
Cn  [cm,  2] 
log  Cm  [cm,  2] 

[nu.2] 
CL,  ['«.*] 
logCm[c,,2] 

log  £.,    =    log-«4 

[rfrf.3] 
log  [rfrf,  3] 

[rfm,  3] 
log  [rfm,  3] 

[rf,.  3] 
log  [rf»,  3] 

[mm,  3] 
Dm  [rfm,  3] 
og  /)„  [rfm,  3] 

[m*,  3] 

Dm  [rf<.  3] 

log  />.  [rfj,  3] 

M  = 

[mm.  4] 

[m,.  4] 

104  METHOD  OF  LEAST  SQUARES. 

The  logarithms  of  the  quantities  in  the  first  row  of  each 
compartment  are  also  written  in,  and  from  these  by  proper 
subtractions  are  obtained  the  logarithms  in  the  margin,  where 

[aft] 
[aa]' 


Ac    = 

£c  = 

[ac] 

...  Am 
...  Bm 

[am] 

[aa]' 
[ftc,  1] 

[ftw»,  1] 

U>t>,  ij 

-  [ftft,  1] 

Now  in  each  compartment  adding  the  logarithms  at  the 
margin  to  each  of  the  logarithms  in  the  first  row  of  that  com- 
partment we  obtain  the  corresponding  logarithms  written  in 
the  other  rows.  The  numbers  represented  by  these  logarithms 
are  next  written  above  them,  and  if  each  of  these  quantities 
is  then  subtracted  from  the  one  above  it  the  result  will  be 
the  corresponding  quantity  in  the  compartment  below.  Some 
of  the  squares  in  each  compartment  are  left  vacant  as  the 
quantities  belonging  to  them  have  already  appeared  above. 

111.  Application  of  Checks.  Also,  by  (93)  and  (94),  in 
the  first  compartment  the  quantities  in  the  first  lines  of  the 
last  column  should  be  equal  to  the  sum  of  all  the  quantities 
in  the  first  lines  of  the  corresponding  rows  plus  the  quantities 
similarly  situated  above  the  first  terms  of  the  rows.  Similar 
checks  will  apply  in  each  compartment;  for  if  from  the 
second  of  equations  (94)  we  subtract  the  product  of  the  first 
equation  multiplied  by  AI,  we  have 

[W,  1]  +  [ftc,  1]  +  .  .  .  [fon,  1]    =    [ft.,  1]     (97) 

In  the  same  manner  we  may  show  that  a  corresponding 
check  holds  throughout,  so  that  finally  we  shall  have 

[mm,  4]    =    [ma,  4]  (98) 


GAUSS'S  METHOD  OF  SUBSTITUTION. 


105 


The  last  compartment  of  the  table  is  added  to  give  this 
final  check  and  the  value  of  [yy]  in  accordance  with  equa- 
tion (96). 

If  the  multiplications  and  divisions  are  simple  or  if  a  table 
of  squares  or  products  or  a  computing  machine  is  used  the 
logarithms  will  of  course  be  omitted  from  the  scheme  of 
solution. 

112.  Example.  In  order  to  illustrate  the  systematic  for- 
mation of  the  coefficients  that  appear  in  the  Normal  Equations 
as  well  as  the  solution  of  the  latter  by  the  above  method  we 
will  take  the 

OBSERVATION    EQUATIONS 


—    Z 


—  2z2  — 


=  0.1 


=        0.4 


First  form  a  table  containing  the  coefficients  in  these  equa- 
tions and  also  the  sums  s.  As  a  first  check  the  sum  of  the 
quantities  in  column  s  should  be  equal  to  the  sum  of  all  the 
quantities  in  all  the  other  columns. 

I.  COEFFICIENTS  IN  THE  OBSERVATION  EQUATIONS. 


No. 

a 

b 

c 

d 

m 

s 

1 

_  i 

1 

1 

1 

—  .1 

1.9 

2 

1 

1 

-  1 

-  1 

-  .6 

—    .6 

3 

1 

—  2 

-  1 

1 

—  .1 

-  1.1 

4 

1 

-  1 

0 

-  2 

-  .3 

-  2.3 

5 

0 

1 

_  1 

1 

.1 

1.1 

6 

1 

0 

-  1 

0 

-  .4 

^4 

Sum. 

3 

0 

-  3 

0 

-  1.4 

—  1.4 

106 


METHOD  OF  LEAST  SQUARES. 


From  these  we  now  compute  the  coefficients  in  the  Normal 
Equations  (B),  and  also  the  necessary  quantities  for  the 
check  equations  (93)  and  (94). 


II.  COEFFICIENTS  IN  THE  NORMAL  EQUATIONS. 


No. 

aa 

ab 

ac 

ad 

am 

as 

bb 

be 

bd 

bm 

i 

2 

1 

1 

-  1 

-  1 

-.6 

-    .6 

1 

-1 

i 
-1 

-.6 

3 

1 

-2 

-1 

1 

-.1 

-1.1 

4 

2 

-2 

.2 

4 

1 

-1 

0 

-2 

-.3 

-2.3 

1 

0 

2 

.3 

5 

0 

0 

0 

0 

.0 

.0 

1 

-1 

1 

.1 

6 

1 

0 

-1 

0 

-.4 

-   .4 

0 

0 

0 

.0 

Sum 

5 

-3 

-4 

-3 

-1.3 

-6.3 

8 

1 

1 

-.1 

No. 

bs 

cc 

cd 

cm 

cs 

dd 

dm 

ds 

mm 

ms 

1 

1.9 

1 

1 

-.1 

1.9 

1 

-.1 

1.9 

.01 

-.19 

2 

-    .6 

1 

1 

.6 

.6 

1 

.6 

.6 

.36 

.36 

3 

2.2 

1 

-1 

.1 

1.1 

1 

-.1 

-1.1 

.01 

.11 

4 

2.3 

0 

0 

.0 

.0 

4 

.6 

4.6 

.09 

.69 

5 

1.1 

1 

-1 

-.1 

-1.1 

1 

.1 

1.1 

.01 

.11 

6 

.0 

1 

0 

.4 

.4 

0 

.0 

.0 

.16 

.16 

Sum 

6.9 

5 

0 

.9 

2.9 

8 

1.1 

7.1 

.64 

1.24 

If  the  coefficients  in  the  Observation  Equations  are  large, 
so  that  it  becomes  convenient  to  use  logarithms,  other  tables 
corresponding  to  I  and  II  would  be  formed  to  contain  these 
logarithms. 


GAUSS'S  METHOD  OF  SUBSTITUTION.  107 

Substituting  these  quantities  now  in  the  general  tabular 
scheme  of  paragraph  110  we  have  the  results  on  the  following 
page.  The  work  of  the  first  compartment  is  performed 
without  the  use  of  logarithms,  as  the  numbers  are  simple. 
The  quantities  in  the  last  column  should  all  be  zero  accord- 
ing to  the  check  equations,  what  small  differences  there  are 
being  due  to  the  rejection  of  figures  beyond  the  third  place 
in  the  decimals.  The  decimal  points  in  logarithms  to  which 
correspond  negative  numbers  have  been  replaced  by  the 
letter  n. 

The  demonstrations  that  have  been  made  now  enable  us  to 
see  at  once  from  an  inspection  of  the  results  in  this  table 
that 

z^    =    0.238 

pZ4    =    1.6 
[wv]    =      .007 

Therefore  substituting  in  equations  (73)  and  (74)  we 
have 

p.Z4    =    .047  r^    —    .032 

If  the  two  values  of  [uu]  obtained  in  the  solution  had 
differed  at  all  we  should  have  taken  the  mean  of  the  two. 

113.  If  the  Elimination  Equations  (95)  are  divided  by 
[aa],  [££>,  1],  [cc,  2],  [cfo?,  3],  respectively,  they  become 


-f  Abz2  -\-  Aczz  -f-  Adz4  -f  Am    =  0 

*a    +    -#c«3    +    -#rf*4    +    J*m     =  0 

Cm   =  0 

J>m     =  0 


And  the  solution  for  the  unknowns  can  be  effected  most 
conveniently  by  arranging  the  computations  in  the  manner 
illustrated  on  page  109. 


108 


METHOD  OF  LEAST  SQUARES. 


SCHEME  A.     SOLUTION  OF  THE  NORMAL  EQUATIONS. 


a 

b              c              d            m             s           8 

5 

-3 

-4 

-3 

-1.3 

-6.3 

0 

-.6 
-.8 
-.6 
-.26 

8 
1.8 

1 
2.4 

1 
1.8 

-    .1 

.78 

6.9 

3.78 

0 
~0~ 

"o" 

5 
3.2 

0 
2.4 

.9 
1.04 

2.9 
5.04 

8 
1.8 

1.1 

.78 

7.1 

3.78 

.64 
.34 

1.24 
1.64 

0 

9n3537 
9B1107 
9n1521 

6.2 
0.7924 

-1.4 

On1461 

-    .8 
9n9031 

-    .88 
9n9445 

3.12 
0.4924 

0 
~0~ 

1.8 
.316 
9.4998 

-2.4 
.181 
9.2568 

-    .14 

.199 
9.2982 

-2.14 

-    .705 
9n8479 

6.2 
.103 
9.0138 

.32 
.114 

9.0552 

3.32 
-    .403 
9n6049 

0 

~b~ 

.30 
.125 
9.0966 

-    .40 
-    .443 
9n6463 

On2403 
9n3587 

1.484 
0.1715 

-2.581 
On4118 

-    .339 
9n5302 

-1.435 
On1569 

i 
T 

T 

6.097 
4.489 
0.6521 

.206 
.589 
9.7705 

3.373 
2.496 
0.3972 

.175 

.077 

8.8889 

.043 

.328 
9.5156 

9M3769 

=  Iog-s4 

1.608 
0.2063 

-    .383 
9n5832 

1.227 

0.0888 

2 

T 

.098 
.091 
8.9601 

-    .285 
-    .292 
9n4657 

0.238 

=  z4            [uv]  =  .007 

.007 

.007 

0 

d 


d 


m 


d 


d 


GAUSS'S  METHOD  OF  SUBSTITUTION.  109 

SCHEME  B. —  SOLUTION  OF  THE  ELIMINATION  EQUATIONS. 


- 


Iogz4 


Iogz2 


log  A 


log  Cdzt 
log  Ad  z^ 


114.    Filling  out  this  table  for  the  example  just  solved,  we 
have 


.238 

.228 

.142 

.260 

.414 

.031 

.143 

.145 

.514 

.191 

.238 

.642 

.318 

1.108 

9.3769 

9.8075 

9.5024 

On2403 

9B3537 

9«7782 

9B1107 

9W9031 

9B7782 

9n2806 

9n1612 

9n6172 

81  U7ft 

9n7106 

n4O  1  D 

9B1551 

110  METHOD  OF  LEAST  SQUARES. 

115.  The  Weights  of  the  Unknowns.  In  order  to  deter- 
mine the  precision  measures  of  zt,  z2,  and  23,  it  would 
next  be  necessary  to  compute  the  weights  of  the  latter  quan- 
tities. The  demonstration  of  the  processes  by  which  these 
weights  may  be  found  will  not  be  taken  up  here,  as  the  best 
method  to  adopt  varies  a  good  deal  with  the  character  of  the 
example,  but  a  statement  of  the  results  in  the  general  form 
of  solution  will  be  given. 

By  treating  the  Elimination  Equations  in  a  way  similar  to 
that  used  in  deriving  equation  (E)  of  paragraph  76  from 
equations  (B)  of  the  same  paragraph,  we  may  show  that 

Zl   +    Am   -f    Bm  a,   -f-    Cm  a,   +    Dm  as     =     0 

*,  +  3m       +   Cm&  +  Dmfr   =    0  (100) 

*•  +   Cm       +  A,  73    =    0 

where  the    a's,    /?'s,    y's,    are  determined  from  the  equations 


a,  =  0     Bd  +  Cdh  +  h  =  Q 

Ac  +^c0l  +  a,   =    0  J?c+&   =    0  (101) 

Ab  +  ai  =  0  Cd  +  73  =  0 

Then  by  an  application  of  the  principles  of  Rule  1,  para- 
graph (77),  it  may  be  shown  that 


1 

Is 
I        dl 

a22                asa 

p\ 

[««]       [&&>  1 

I 

]  "  "  [cc,  2]    "  [rfrf,  3] 

v\ 

[fed,  1 

1  +  [cc,  2]  +  [dd,  3] 

i          "  V     (102) 

[cc,  2]        [cirf,  3] 

1 

[dd,  3] 


GAUSS'S  METHOD  OF  SUBSTITUTION.  HI 

These  equations  can  of  course  be  extended  to  cover  any 
number  of  unknown  quantities,  and  tabular  schemes  for  the 
computations  of  the  a's,  /?'s,  y's,  .  .  .  and  of  the  weights 
Pzj  Pzz>  Pzz,  ...  can  readily  be  arranged. 

For  a  general  demonstration  of  these  results,  and  also  for 
a  discussion  of  special  methods  of  solution,  consult  — 

Johnson,  "The  Theory  of  Errors  and   Method  of  Least  Squares," 

chap.  ix. 

Wright,  "  Treatise  on  the  Adjustment  of  Observations,"  chap.  iv. 
Chauvenet,  "  Spherical  and  Practical  Astronomy,"  pp.  530-649. 


THE    METHOD   OF   CORRELATIVES. 

116.  The  method  of  adjusting  "Conditioned  Observa- 
tions "  explained  in  paragraph  33  is  perfectly  general,  but 
where  there  are  many  conditions  to  be  satisfied  the  solution 
is  apt  to  be  very  laborious.  For  the  case  that  occurs  most  fre- 
quently in  practice,  in  which  the  observations  are  direct  and 
equal  in  number  to  the  number  of  unknown  quantities,  the 
process  of  solution  devised  by  Gauss  and  called  the  "  Method 
of  Correlatives"  is  the  most  convenient.  This  method  is 
derived  as  follows  :  — 

Let  q  observations,  M^  M^  .  .  .  Mq,  of  the  respective 
weights  />!,  p.,,,  .  .  .  pq,  be  made  directly  upon  the  values 
of  q  unknown  quantities,  and  let  the  most  probable  values 
of  the  unknowns  be 


=    J/i  -f  VH      za    =    J/a  -f  »„ 


q. 


Where  v1?  u2,  .  .  .  vq,  are  the  most  probable  corrections  to 
apply  to  the  observed  values  as  well  as  in  this  case  the 
residuals  of  the  observations. 

If  the  ri  condition  equations  are  not  linear  they  may  be 
reduced  to  that  form  by  the  method  of  paragraph  44,  so  that 
we  may  assume  for  our 


112  METHOD   OF  LEAST  SQUARES. 

CONDITION    EQUATIONS 

«l"l    +    a2«2    +    •    •   •   aqVq    +    ™-\      —       0 

blvl  -f-  £2t?2  -f-  .  .  .  £?u9  +  w*2    =    0 

(A) 


In  which  the  quantities  mlt  w2,  .  .  .  mn',  would  all  be  zero 
if  the  observations  were  exact.  It  is  to  be  observed  that  the 
coefficients  a,  d,  .  .  .  £  are  not  arranged  in  the  same  order  in 
these  equations  as  they  are  in  the  observation  equations  of 
paragraph  107. 

The  values  of  vt,  «2,  •  •  •  vq,  must  be  determined  so  as  to 
satisfy  the  above  equations  and  also  by  the  principle  of  Least 
Squares,  so  as  to  make 


-f-  Pz°^  +  •  •  •  Pqvq     =•    a  minimum. 
Corresponding  with  a  minimum  value  of  this  last  we  have 

PlVldOl    +  P2V2dv2    -f    .  .   .  pqVqdvq      =      0  (B) 

for  all  possible  simultaneous  values  of   do^  c?y2,  .  .  .  dvq  ;  that 
is,  for  all  values  which  satisfy  the  equations, 


-f-  a2dv2  -|~  .  .  .  aqdvq    =    0 

=    0 

(C) 


lqdvq    =    0 


lidVi  -J-   lzdvz  -f-   •  •  •    Iqdvq    =    0 

obtained  by  differentiating  equations  (A). 

Therefore,  denoting  the  first  member  of  (B)  by  R  and  the 
first  members  of  (C)  by  S^  S2,  ...  £„,>,  it  will  be  nec- 
essary that 

R  _  k^  -  k2Sz-  ...  kn,Sn,    =    0  (D) 

where   &t,  k%,  ...  kn<,    are  undetermined  coefficients. 


GAUSS'S  METHOD  OF  SUBSTITUTION.  H3 

This  last  equation  will  be  satisfied  if  the  coefficient  of  each 
differential  in  it  is  made  equal  to  zero,  that  is,  if 


(103) 
Pqvg    ==    kvaq  -f-  KyOq  -(-•••  kn'lq 

All  that  remains  therefore  is  to  find  values  of  viy  v2,  ...  vq 
and  A^,  A;2>  .  .  .  AV,  which  will  satisfy  simultaneously  equa- 
tions (A)  and  (103),  and  that  this  may  be  done  is  easily 
seen  from  the  fact  that  we  have  the  same  number  of  equations 
as  unknowns. 

Substituting  the  values  of  wx,  v2,  ...  vq  from  equations 
(103)  inequations  (A)  we  have  the  following : 

CT«  ab  al 

MZ-—     -  #2i.—  *vZ  — 

jr.    y    <tf>  ,,    -  &&  ,.      y  ^^ 

to  *  -p-  '  -  *fe  *  *T     -•••«»' 2  p-  ' 

(104) 

CT?  6?  II 

The  solution  of  these  equations  gives  at  once  the  values  of 
A*,,  &2,  .  .  .  Ay,  which  are  called  the  "Correlatives"  of  the 
Condition  Equations.  The  values  of  Vj,  u2,  .  .  .  vq  are  then 
found  by  substituting  the  values  of  the  k's  in  equation  (103). 

117.  As  equations  (104)  are  of  the  same  general  form  as 
a  set  of  Normal  Equations,  Gauss's  Method  of  Substitution 
can  be  advantageously  employed  in  the  solution. 


114  METHOD  OF  LEAST  SQUARES, 

When  there  is  but  a  single  equation  of  condition  the  second 
members  of  equations  (103^)  reduce  to  their  first  terms,  and 
equations  (104)  reduce  to  the  single  equation 

CLd 

^2—  +  m,    =    0  (105) 

and  the  values  of  vlt  v2y  ...  vq   in  (103)  become 


a, 

v    = 


aa 
'  P 

It  is  from  these  results  that  the  rules  in  paragraph  35   are 
derived. 

118.    Example.      Suppose  we  have  given  the  observations 

MI  =  2.02,  weight  3 

Mt  =  4.13,              «       2 

Ms  =  2.52,              "       5                        (a) 

MI  =  2.67,              «       7 

M&  =  2.84,              »       4 

and  let  the  most  probable  values  of  the  unknowns  be  repre- 
sented by 

Z,    =    Mi  +  «!,       22    =    M2  -\-  Vt,    .  .   .    25    =    JJf5  +  VS     (b) 

Also  suppose  that  the  unknowns  are  subject  to  the  conditions 

Z  2  2  S  2        —      14-0 


-3.  =         1.5 


(c) 


GAUSS'S  METHOD  OF  SUBSTITUTION. 


115 


Then  expressing  these  conditions  in  terms  of  the  corrections 
by  means  of    (a)    and    (b),    we  have  the 


CONDITION    EQUATIONS 

^1    +     V».     +     V3    +     U*    +     U5    +     -18       =       0 

w2  -  vt  -   .04    =    0 

Referring  to  paragraph   116,   we  see  that  in  this  example 
n'    =    2,         m1    =    0.18,         mz    =     —  0.04 

For  the  purpose  of  computing  the  coefficients  in  equations 
(104)    we  next  arrange  the  following  table. 


p 

a 

b 

aa 

ab 

Ib 

P 

P 

P 

1 

3 

1 

0 

•  — 

0 

0 

3 

1 

1 

1 

2 

1 

1 

2 

¥ 

2 

1 

5 

1 

0 

0 

0 

IT 

i 

i 

1 

7 

1 

-  1 

—  -  — 

7 

7 

7 

1 

4 

1 

0 

0 

0 

4 

599 

5 

9 

420 

14 

14 

116  METHOD  OF  LEAST  SQUARES. 


Substituting  these  results  in  equations   (104),  we  have 

—  ^  4-  —  &2  4-  .18    =    0 
420  14 

(e) 

5  9 

—  k,  -\ >L  —  .04    =    0 

14  14 

Solving,  &!    =   —  .1647  (^ 

&2    =        .1537 

Then  from  equations   (103)   we  get  at  once 

t>!    =    —  .0549,         v4    =    —  .0455, 

v2    =    —  .0055,         vs    =    —  .0412.  (g) 

vs    =    —.0329, 

And  by  substituting  these  results  in    (b)  we  can  obtain  the 
values  of   z^  22,  28,  24,  z6. 

The  above  is  the  solution  of  Example  70,  page  127. 


EXAMPLES. 


1.  An  urn  contains  five  black  balls,  three  red  balls  and 
two  white  balls.     If  three  balls  are  drawn  from  the  urn  what 
different  combinations  may  result,  and  what  is  the  probability 
of  each  ? 

2.  In  a   single  throw  with   a  pair  of  dice  what  is   the 

§ 
probability  that  neither  ace  nor  doublets  will  appear  ? 

y 

3.  Four   cards  are   drawn   from  a  pack.     What   is   the 
probability  of  getting  four  aces?     Of  getting  one    of   each 
suit? 

4.  From  a  lottery  of  thirty  tickets,  marked  1,  2,  ...  30, 
four  tickets  are  drawn.     What  is  the  probability  that  num- 

2 
bers  1  and  15  will  be  among  them? 

145 

5.  Find  the  odds  against  the  appearance  of  7  or  11  in  a 
single  throw  with  a  pair  of  dice.  7  :  2 

6.  I  toss  up  n  coins.     What  is  my  chance  of  getting  just 
one  head  ? 

7.  In   a  single  throw  what  are  the  relative   chances  of 
throwing  9  with  two  dice  and  with  three  dice  ?  24  :  25 

8.  From  2  n  counters  marked  with  consecutive  numbers 
two  are  drawn.     What  are  the  odds  against  having  an  even 
sum  ?  n  :  n  —  1 

9.  In  two  trials  with  a  single  die  what  is  the  probability 
of  throwing  (a)  an  ace  the  first  time  only?    (b)  at  least  one 
ace  ? 

10.  Find  the  probability  of   throwing   doublets    one    or 

91 
more  times  in  three  trials  with  a  pair  of  dice. 

216 


118  METHOD   OF  LEAST  SQUARES. 

11.  Find  the  probability  of  throwing  exactly  three  aces 

125 

in  five  trials  with  a  single  die.  

3888 

12.  A  certain  stake  is  to  be  won  by  the  first  person  who 
throws  5  with  a  die  of  twelve  faces.     What  is  the  chance  of 
the  sixth  person  ? 

13.  A  and  B  play  chess.     A  wins  on   the  average  two 
games  out  of  three.     What  is  A's  chance  of  winning  just 

80 
four  games  out  of  the  first  six  ? 

243 

14.  A  and  B  shoot  alternately  at  a  mark.     A  hits  once  in 
n  times  and  B  once  in  n  —  1  times.     Find  their  chances  of 
first  hit,  and  the  odds  in  favor  of  B  if  A  misses  on  his  first 
shot.  Even,     n  :  n  —  2 

15.  In  how  many  trials  will  it  be  a  wager  of  4  to  3  that 
double  five  will  be  thrown  with  a  pair  of  dice  ?  30 

16.  Find  the  probability  of  throwing  one  and  only  one 

5 
ace  in  two  trials  with  a  single  die. 

18 

17.  If  I  have  three  tickets  in  a  lottery  of  four  prizes  and 

41 

eight  blanks,  what  is  my  chance  of  drawing  a  prize  ? 

55 

18.  Find  the  probability  of  throwing  at  least  four  aces  in 

203 
six  trials  with  a  single  die.  

23328 

19.  On  an  average  seven  ships  out  of  eight  return  to  port. 
Find  the  chance  that  out  of  five  ships  expected  at  least  three 

...  16121 

will  return. 

16384 

20.  In  a  lottery  containing  a  large  number  of  tickets, 
where  the  prizes  are  to  the  blanks  as  1  :  9,  find  the  chance 

of  drawing  at  least  two  prizes  in  five  trials.  

100000 


EXAMPLES.  119 

21.  In  a  purse  are  ten  coins,  all  nickels  except  one  which 
is  a  five-dollar  gold  piece ;    in    another   are   ten   coins,  all 
nickels.     Nine   coins   are   taken   from   the   first   purse   and 
placed  in  the  second,  and  then  nine  coins  are  taken  from  the 
latter  and  placed  in  the  former.     If  you  now  had  your  choice 
which  purse  would  you  take  ? 

22.  A  and  B  engage  in  a  game  in  which  A's  skill  is  to 
B's  as   2:3.     What  is   A's   chance  of  winning  at  least  two 
games  out  of  five  ? 

23.  If  A's   skill  at  a  certain  game  is  double  that  of   B, 
what  are  the  odds  against  A's  winning  four  games  before  B 
wins  two?  131  :  112 

24.  A  party  of  twenty-five  take  seats  at  a  round  table. 
What  are  the  odds  against  any  two  specified  persons  sitting 
next  to  each  other  ? 

25.  A  has  three  shares  in  a  lottery  in  which  there  are  three 
prizes  and  six  blanks.     B  has  one  share  in  another  where 
there   is   but   one    prize  and  two  blanks.     What  are   their 
relative  chances  of  getting  a  prize  ?  A  :  B  =  16  :  7 

26.  Expand  through  the  terms  involving  h*  and   &8,  the 
expression 

-  +  (y  +  *)' 

'X   -\-    h 

When    a;    is   1    and    y    is    -,    does    -  4-  y*   increase   or 

2  x 

diminish  when  x  and  y  begin  to  increase  at  the  same  rate? 

27.  Given  f  (x,  y)    =    x2(a  -j-  y)8,   expand  the  expres- 
sion   (a;  4-  A)2(a  +  y  +  &)'. 

28.  Find  the  value  of   Iog10  a  -j-  cos  ft,  when    a  •=.  1001 
and    b  =  0.1°.     Give  the  result  first  to  five  places  and  then 
to  seven  places  of  significant  figures,  in  each  case  without  the 
aid  of  tables.  4.000433 

29.  Transform  to  the  new  origin,    (2,  —  3,  1),    the  equa- 
tion, z2  -f  </2  -4-  a2  —  4  x  -f-  6  y  -  2  z  -  11  =  0,   the 


120  METHOD  OF  LEAST  SQUARES. 

axes  remaining  parallel  to  the  original  ones.  The  equations 
of  transformation  are  x  =  2-j-a;',  y  =  —  3-j-t/', 
2=1  +  2'.  x2  -f  y2  4-  22  =  25 

30.  Find  the  minimum  value  of  »2  -)-  xy  -|-  y2  —  a«  —  by. 

(ab  —  a*  —  b2) 
3 

31.  Find  the  values  of  «,  y  and  2   that  render  a  maximum 
or  a  minimum  the  function    a;2  -f-  y2  -J-  z2  -(-  a;  —  2  2  —  a?y. 

32.  Find  the  values  of   x   and   y   that  render  a  maximum 
or  a  minimum  the  expression   sin  x  -\-  sin  y  -j-  cos  (a;  -j-  yj. 

33.  Find  the  co-ordinates  of  a  point  the  sum  of  the  squares 
of  whose  distances  from  three  given  points,    (a^,  y, ),  (z2,  y2), 

.  .      .  (Xl   -|-   «2   +   «3) 

(a:8,  y8),    is  a  minimum.  a;  =   - 

o 

34.  Given  the  volume,  a8,  of  a  rectangular  parallelepiped, 
find  its  shape  when  its  surface  is  a  minimum. 

35.  Find  the  volume  of  the  greatest  rectangular  parallelo- 
piped  that  can  be  inscribed  in  the  ellipsoid 

a^         y2         2«  Sabc 

^ " "  *• " "  ^  :  ~3~7^ 

36.  In  the  Physical  Laboratory  apparatus  for  illustrating 
the  Estimation  of  Tenths,  the  reading  of  the  micrometer  head 
for  a  certain  setting  is  computed  to  be  2.3038,  which  may  be 
taken  as  exact.     Setting  the  apparatus  by  the  eye  the  follow- 
ing readings  are  obtained  :  — 

2.314         2.324         2.310         2.519         2.326 
2.320         2.302         2.313         2.305 

What  are  the  accidental  and  real  errors  and  the  residuals  ? 
Are  there  any  constant  errors  or  mistakes  ?  If  there  are 
constant  errors  are  they  of  the  first,  second,  or  third  class  ? 
How  do  you  tell? 


EXAMPLES.  121 

37.  Are  the  following  observations  such  as  to  call  for  an 
application  of  the  Method  of  Least  Squares  in  their  adjust- 
ment? 

x  -\-    y  —    z  -\-  u   =         5  3«  —  y  —    z  —    u   =    1 

x  —  2y  -\-2z-\-  u   =    —  1  x  —  y  -f-  4z  -|-  6w    =    9 

38.  In  the  case  of  direct  observations,  what  other  quanti- 
ties  besides    the    arithmetical    mean    might    reasonably   be 
assumed  to  give  plausible  values  of  the  unknowns  ?     Why  is 
the  arithmetical  mean  preferred  to  these  ? 

39.  Find  the  most  probable  value  of  a  quantity   M  from 
the  observations 

216.27         216.16         216.04         216.19         216.44        216.58 
.29  .43         215.99  .39  .51 

.33  .09         216.23  .14         215.94 

Also  test  the  result  by  finding  the  sum  of  the  residuals. 

M  —   216.251 

40.  In  the  determination  of  a  certain  wave  length,  Row- 
land made  the  following  observations.     Find  the  most  prob- 
able wave  length. 

4.524  4.515  4.513  4.507  4.501  4.485  4.517  4.493  4.505 
.500     .508     .511     .497     .502     .519     .504     .492 

4.5055    ±  0.0017 

41.  Ten  measurements    of  the  density  of   a   body  gave 
results  as  follows.     Find  the  most  probable  density. 

9.662    9.664    9.677    9.663    9.645 
.673     .659     .662     .680     .654 

9.6639  ±  0.0022 

42.  In  a  triangulation  of  the  U.  S.  Coast  Survey  an  angle 
was  measured  twenty-four  times  witli  results 


122  METHOD   OF  LEAST  SQUARES. 

116C 


43' 

44 

".45 

48" 

.90 

47".40 

47" 

.85 

51 

".75 

50 

.55 

49 

.20 

47  .75 

50 

.60 

49 

.00 

50 

.95 

48 

.85 

51  .05 

48 

.45 

52 

.35 

51 

".30 

49".05 

46 

".75 

51 

.05 

50  .55 

49 

.25 

51 

.70 

49  .25 

53 

.40 

Find  the  best  value  for  the  angle.         116°  43'  49".64  ±  0".28 

43.  The  following  observations  were  made  with  a  sextant 
in  order  to  determine  the  latitude  of  a  place  :  — 

43°  4'  46"         7",     59" ,     52"       47"       36" 
24        28        39        52         15        40 

What  is  the  most  probable  latitude  ?  43°  4' 1*7"  ±  5" 

44.  Determine  the  quantity   M   from  the  observations 

M  =   81.55    .41    .68    .25   ..27    .77    .13    .86    .10    .03 
Wts.  25     25     16       9       9       9       4       1       1       1 

45.  An  angle   M  is  measured  with  the  following  results : 

M  p  M  p         M  p         M  p         M  p 

45".00  5  42".50  5  27".50  3  36".25  2  45".00  2 

31  .25  4  37  .50  3  43  .33  3  42  .50  3  40  .83  3 

45  .00  3  38  .33  3  40  .63  4  39  .17  3 

Find  the  most  probable  value  of  the  angle  and  test  the 
result  by  computing    2/?u.  39".78  ±  0".94 

46.  In  one  hundred  measurements  of  angles  made  in  the 
primary  triangulation  in  Massachusetts  by  the  Coast  Survey 
there  were  found  between 

6".0    and   5".0       1  error.  1".0    and      0".0  26  errors. 

5  .0      "      4  .0       2  errors.  0  .0      "    —  1  .0  26      " 

4  .0      "     3  .0       2      «  —  1  .0      "    —  2  .0  17      " 

3  .0      "     2  .0       3      "  —  2  .0      "    —  3  .0       8      " 

2  .0      "      1  .0     13      "  —  3  .0      "   —  4  .0       2      " 

Plot  these  results  as  in  Figure  1,  page  8. 


EXAMPLES.  123 

47.  In  sixty-six  determinations  of  the  velocity  of  light 
made  at  Washington  the  percentage  of  errors  of  different 
magnitudes  was  found  to  be  as  follows:  — 


Over 
Equal  to 

-  7.0 

-7.0 
-6.0 

.8% 
.8 
1.6 

Over          7.0 
Equal  to  7.0 
6.0 

0.0% 
1.6 
3.5 

-5.0 

2.0 

5.0 

4.2 

-4.0 

3.9 

4.0 

5.8 

—  3.0 

7.8 

3.0 

8.6 

-2.0 

12.8 

2.0 

12.4 

—  1.0 

17.1 

1.0 

17.1 

Plot  the  results  and  draw  a  smooth  curve. 

48.  Given  the  observations 

x  -       y  +  2z    =    3  4a  -|-     y  -f-  43    =    21 

3;c  -f-  2y  —  52    =    5       —  x  -j-  3y  -f-  3z    =    14 

find  the  most  probable  values  of   x,   y,   and   z. 

x    =    2.470  ±  .038 

49.  0  being  the  level  of  the  sea  and   flt  _P2,  P&->   three 
points  whose  altitudes  are  to  be  determined,  the  following 
observations  are  made:  — 

Pl  above  0    =    10  ft.         P2  above  P3    =    9^  ft. 
P2       «      PL  =      7  Pl      "       P8    =    2 

P2       "      (9    =    18 

Find  the  most  probable  altitudes.  Ps    =    8.50  ±  .29 

50.  The  altitudes  of   A   above    0,    H   above   A,   and  JS 
above    O  are   found   by    measurements   to   be   respectively 
12.3,  14.1,  and  27.0  feet.     What  is  the  most  probable  value 
of  each  of  these  differences  in  level?         A   =    12.50  ±  .17 

51.  Measurement  of  the  ordinates  of  points  on  a  straight 
line  corresponding  to   abscissas  4,  6,  8,  9,  are   made    with 
results  5,  8,  10,  12.     What  is  the  most  probable  equation  of 
the  line  in  the  form   y  =   mx  -\-  bf  b  =    —  0.29 


124  METHOD  OF  LEAST  SQUARES. 

52.  Find  the  altitudes  in  Example  49  if  the  observations 
have  weights  5,  3,  6,  2,  4,  respectively.  -" 

53.  Solve  the  example  in  paragraph  31,  giving  the  obser- 
vations the  weights  25,  25,  4,  4,  4,  4,  4,  4,  1,  respectively. 

Elevation  of   P5   =   320.25 

54.  Find  the  most  probable  values  of   zt,  22,  and  z8  from 
the  observations 

2X  =  552.10  wt.  16  21  —  22   =         -75  wt.  1 

—  z2  -f-  za  =         .15    "      9     Zi  -J-  z2  —  zs  =   552.05    "    1 

z3  =  551.23    "4  zl  —  z3  =         .70    "    1 

22  =  551.30    "      4 

22  =   551.2345 

55.  In  the    triangulation    of  Lake  Superior   there  were 
measured  at  station  0  the  angles 

F  0  P  =     62°  59'  40".33  wt.  5 

F  0  E  =     64  11  34  .92  «    7 

F  0  B  =  100  20  29  .12  "    4 

P  0  B  =     37  20  49  .55  «    7 

E  O  B  =     36  8  55  .86  "4 

Required  the  adjusted  values  of  the  angles. 

F  0  P  =   40".28  ±  0".34 

56.  In  the  U.  S.  Lake  Survey  the  following  angles  were 
measured  at  station  North  Base  :  — 

(1)  Crebassa— Middle                 55°  57'  58".68  wt.    3 

(2)  Middle  — Quaquaming           48    49    13  .64  "    19 

(3)  Crebassa  — Quaquaming      104    47    12  .66  «    17 

(4)  Quaquaming  —  South  Base    54     38    15  .53  "    13 

(5)  Middle  — South  Base            103     27    28.99  «      6 

Find  the  adjusted  values  of  the  angles. 

(1)    =   58".965;    r  =   0".28 


EXAMPLES.  125 

-  57.     Adjust  the  following   observations  of  differences  in 
level: — 

Altitude  of  A    401.3    wt.  16  C  above  B      72.5    wt.  9 

A  above  B    220.8      "    16  A     «       B    222.0     «    1 

A      «       (7    150.2      "      4        Altitude  of  J?    180.7      "    1 


58.     ] 


)8.  In  "Conditioned  Observations"  can  the  number  of 
observations  required  be  less  than  the  number  of  unknown 
quantities?  Why  must  the  number  of  conditions  be  less 
than  the  number  of  unknowns? 

59.  From  the  following  measurements  of  the  angles  formed 
at  the  centre  of  a  disk  by  four  radial  lines,  find  the  most 
probable  values  of  the  angles. 

A    =    104°  25'  13"  O   =    86°  33'  20" 

B    =      98    13    47  D    =    70    48    23 

A    =    104°  25'  2".25 

Also  solve  giving  the  observations  the  weights  5,  2,  1,  4, 
respectively. 

60.  Four  observations  on  the  angle    A   of  a  triangle  gave 
a  mean  of   36°  25'  47",    two   observations   on    B    gave   a 
mean  of   90°  36'  28",    and  three  on    G  gave    52°  57'  57". 
Adjust  the  triangle.  A   =   36°  25'  44".2 ;    r  =   7".7 

61.  Five  angles  at  a  station  are  measured,  and  also  their 
sum.     The  observed  sum  differs  from  the  sum  of   the   five 
observed  parts  by  the  amount   d.     What  are   the   adjusted 
values  of  the  angles  ? 

62.  The  three  angles  of  a  spherical  triangle  are  measured 
with  results 

A   =  46°  17'  38".32          B  =   73°  35'  16".15 
C  =   60°  7'  5".16. 

Adjust  the    triangle,  knowing  that  the    spherical    excess  is 
2".475.  A   =   39".3;    ^   =    1".6 


126  METHOD  OF  LEAST  SQUARES. 

63.  At  the  station  Pine  Mountain  the  following  angles 
were  observed  between  surrounding  stations  :  — 

Jocelyne  —  Deepwater  65°  11'  52".500  wt.  3 

Deepwater  —  Deakyne  66  24    15  .553  "    3 

Deakyne  — Burden  87  2    24  .703  "    3 

Burden— Jocelyne  141  21    21  .757  "    1 

Find  the  most  probable  values  of  the  angles. 

64.  Solve  Examples  55  and  56  by  the  method  of  "  Con- 
ditioned  Observations." 

65.  A  is  a  station  whose  altitude  is  known  to  be  5240.1 
feet.     JB  and  C  are  floats  on  a  lake,  and  D  is  a  signal  point. 
From  the  following  observations  determine  the  most  prob- 
able altitudes  of  J?,   C  and  D. 

C  below  A  720.1  wt.  3  B  below  A  719.7  wt.  3 
D  «  A  200.3  "5  B  «  D  520.9  "  2 
C  "  D  520.4  "  2 

66.  Given  the  following  observations,  subject  to  the  con- 
dition    Zi  -j-  za    =    zs,    find   the   most  probable  values  of 
zly  z^  and  z3. 

2zi  —  za  +  zs  =   3.0     222  —  za  =  1.0 
2z!  —  3z2  =  —  4.5      Zi  -f-  222  =  5.1 
Zl  +  z3  =   3.8 

67.  The  chemical  composition  of  a  specimen  was  found 
by  several  observers  to  be  as  follows :  — 

Pb  =  .52      Other  substances     =  .09      Au  and  Ag  =  .39 
Ag  =  .27      Pb  and  Ag  =  .78      Impurities    =  .10 

Au  =  .11      Pb  and  impurities  =  .62      Au  =  .12 

From  these  observations  find  the  most  probable  composition 
of  the  specimen. 


EXAMPLES.  127 

68.  From  the  following  observations  what  are  the  best 
values  of  the  unknowns,  supposing  that   y   and   z   must  be 
equal  ? 

x  -\-  y    =    5.2     wt.  4         y  -\-  z    =    4.2     wt.  I 
x    =    3.0      "    9  z    =    2.0       "  4 

85    --    «     =     1.1         "      1 

69.  In  determining  the  difference  in  longitude  between 
various  cities  the  results  obtained  were 

(1)  Cambridge  —  Washington  23™  4K041  wt.  30 

(2)  Cambridge  — Cleveland  42     14.875  "      7 

(3)  Cambridge  —  Columbus  47     27.713  "      8 

(4)  Washington  —  Columbus  23     46.816  "      7 

(5)  Cleveland  —  Columbus  5     12.929  "      5 

Adjust  these  observations. 

70.  The  capacity  of  a  condenser  is  known  to  be  14.0  m.  f. 
It  is  divided  into  five  sections,  a,  b,  c,  d,  e,  and  it  is  known 
that  the  difference  between    b   and    d  is    1.5  m.  f.     Find  the 
most  probable  capacities  of  the  sections  from  the  observa- 
tions 

a    =    2.02       wt.  3  d   =    2.67       wt.  7 

b    =    4.13        "2  e    =    2.84        "    4 

c    =    2'52        "    5  a    =    1.9651 

71.  If  the  unknowns  in    the  following  observations  are 
subject  to  the  condition    x  -|-  2y  -|-  3z    =    36,    what  are 
their  adjusted  values? 

x  =  4.3     wt.  1,       y  =  5.7     wt.  4,       z  =  7.3     wt.  9 

x    =    3.77 

72.  A  cannon  is  discharged  horizontally  from  the  top  of 
a  bluff.     Observations  on  the  time,  and  distance  of  fall  of  the 
ball  gave  the  results 

t    =    0.5       1.0       1.5         2.0    seconds 
8    =    1.2       4.0       9.1        15.0    metres 


128  METHOD  OF  LEAST  SQUARES. 

What  curve,  passing  through  the  point  of  departure  of  the 
ball,  will  represent  the  above  observations  ? 

73.  An  Argand  burner   shows  the  following  efficiencies 
with  varying  rates  of  gas  consumption  : 

g    =    2.0     2.3     2.8     3.3     4.0     4.5     5.0     feet 
E  =    2.1     2.4     2.5     3.0     3.2     3.8     4.1 

Find  the  equation  of  the  straight  line  which  best  rep- 
resents the  relation  between  g  and  E.  The  measurements 
on  g  are  without  appreciable  error. 

74.  Observations  are  made  upon  the  expansion  of  Amyl 
alcohol  with  change  in  temperature  as  follows :  — 

V  =    1.04       1.12       1.19       1.24       1.27     cu.  cm. 
t    =    13.9       43.0       67.8       89.0       99.2     C.  degrees 

If  V  =  1  -)-  -Z?  t  -j-  C  t2  expresses  the  law  connect- 
ing the  volume  and  temperature,  find  the  most  probable 
values  of  B  and  C. 

75.  In  a  Hooke's  joint  where  the  angle  between  the  axes 
is  45°,    x  being  the  angular  rotation  of  the  driver,  and    y 
that  of  the  follower,  from  the  following  measurements  find 
the   equation    of   a   curve   that  will  represent   the   relation 
between   x   and   y  —  x. 


X 

y  —  x 

x 

y  —  x 

x 

y  —  x 

0° 

o°.o 

80° 

-  5°.8 

140° 

8°.8 

20° 

-  5°.7 

90° 

-  2°.0 

160° 

5°.3 

40° 

—  9°.9 

100° 

2°.3 

180° 

o°.o 

60° 

-  10°.4 

120° 

8°.0 

y  —  x  =    —  0.85  —  9.82  sin  2a  -f  0.92  cos  2a 

76.  A  series  of  observations  extending  over  a  period  of 
thirty  years  was  made  by  Quetelet  to  determine  the  daily 
variation  in  temperature  at  Brussels.  The  mean  results  of 


EXAMPLES.  129 

the  measurements  are  given  below.     From  them  derive  an 
equation  to  express  the  temperature  at  any  time  of  the  year. 

Jan.  4°.66  May      9°,83  Sept.  8°.16 

Feb.  5°.42  June  10°.09  Oct.  6°.55 

Mar.  6°.77  July      9°.71  Nov.  5°.10 

Apr.  8°.59  Aug.     9°.14  Dec.  4°.41 

y  =  7.369  +  0.9854  sinSOz  -  2.7084  cos30x 
-f  0.0100  sin60a;  —  0.1950  cosGOa; 
-  0.0133  sin  90sc-f  0.1783  cos  90a 

In  this  answer  the  values  of   x   begin  at  the  15th  of  Janu- 
ary, and  represent  the  time  in  months. 

77.  The  law  connecting  the  time  of  vibration  of  a  pendu- 
lum with  its  length  is  assumed  to  be  of  the  form,    T  =  m  Ln. 
From  the  following  observations  find  the  most  probable  values 
of   m   and   n. 

T   =      12.9     'll.6       10.4       9.7       5.3       4.6 
'L    =    164.4     132.9     107.6     93.5     28.4     20.6 

L   is  in  centimetres,    T  in  tenths  seconds.         n  —    0.5000 

m   =   1.0044 

78.  Determine  the  equation  of  a  curve  which  will  repre- 
sent the  following  observations : 


X 

0.0 

y 

0.00 

X 

1.5 

y 

1.09 

X 

3.0 

y 

8.65 

0.5 

0.04 

2.0 

2.56 

3.5 

13.72 

1.0 

0.31 

2.5 

4.99 

4.0 

20.47 

79.     Determine  the  equation  of  a  curve  which  will  repre- 
sent the  relation  between    x   and    y    in  the  observations, 


X 

y 

x 

y 

x 

y 

x 

y 

0.0 

4.51 

0.3 

4.09 

0.6 

3.03 

1.2 

0.92 

0.1 

4.44 

0.4 

3.76 

0.8 

2.24 

1.5 

0.38 

0.2 

4.31 

0.5 

3.42 

1.0 

1.49 

2.0 

0.05 

130  METHOD   OF  LEAST  SQUARES, 

80.  At  a  station  P  the  angles  between  a  straight  line 
passing  through  P  parallel  to  the  axis  of  X  and  the  direc- 
tions from  P  of  four  points  P1}  P2,  _P8,  P4,  are  measured. 
Having  given  the  coordinates,  (a,  £>),  of  the  four  points, 
find  the  coordinates  of  P. 


Point. 
Pi 


If  the  coordinates  of  the  point    P    are    (x,  y),    and  the 
angle  is  denoted  by   A,    we  have 


Coordinates. 

Angle. 

a 

6 

4.21 

3.24 

39°   18' 

1.21 

2.10 

147°  54' 

—  0.51 

0.22 

205°  24' 

2.50 

-  1.10 

277°  15'' 

x  —  a 


81.  If   a  sin  bx  =   M,    and  values  of   M  are   observed 
for  known  values  of   a   and   £>,   determine  the  most  probable 
value  of   x.     If   x'   is  an  approximate  value  of  x  found  by 
trial,  and    m   =   a  sin  bx'  —  M,    we  shall  have 

V  2  a  b  m  cos  bx' 

/j#        —  .   -         /v«'         _  .  _ 

~2t(a  b  cos  bx')'2' 

82.  If  in  one  series  of  observations  the  value  of    h   is 
twice  what  it  is  in  another,  what  is  the  relative  probability  of 
the  occurrence  of  an  error  of  given  magnitude   a  in  the  two 
series  ?     Show  what  the  curves  of  error  will  be  in  the  two 
cases.    What  error  has  the  same  probability  for  its  occurrence 
in  each  series  ?    What  is  the  relative  probability  of  the  occur- 
rence of  an  error  not  greater  than  a  in  the  first  case  and  not 
greater  than   2a   in  the  second  case  ? 

83.  From  64  observations  the  latitude  of  a  station  was 
found  to  be   49°  10'  9".110  ±  0".051.     What  was  the  prob- 
able error  of  a  single  observation  ?  0".41 


EXAMPLES.  131 

84.  If  twenty  measurements  of  an  angle  give  a  result  with 
an   A.D.   of   0".38,    and  it  is  required  to  find  the  angle  so 
that  the   A.D.    shall  be  only  0".25,  how  many  more  observa- 
tions must  be  made  ?  27 

85.  From  the  following  determinations  of  the  area  of  a 
field  find  the  most  probable  area  and  its  probable  -error. 


5674  ±  12,  5680  ±  4,  5685  ±  3,  5682  ±  1,  5678  ±  2 

4  =  5681.41  ±  0.84 

86.  From  the  following  measurements  by  Fizeau   and 
x  others,  find  the  most  probable  value  for  the  velocity  of  light 

together  with  its  probable  error.     Measurements  are  in  kilo- 
meters. 

298000  ±  1000         299990  ±     200         299930  ±  100 
298500  ±  1000         300100  ±  1000 

V  =    299917  ±  88 

87.  Two  different  instruments  give  for  the  value  of  an 
angle,  f 

11  * 

34°  55'  33".0  ±  4".l,  34°  55'  36".0  ±  6".3 

What  is  the  best  value  to  take  for  the  angle  ? 

34°  55'  33".9  ±  3".4 

88.  Determinations  of  the  difference  in  longitude  between 
Washington   and  Key  West  made  on  seven  different   days 
gave  the  results 

I9m     1'.42  ±  0'.044  19m     ls.60  ±  0'.046 

1  .37  ±  0  .037  1  .55  ±  0  .045 

1  .38  ±  0  .036  1  .57  ±  0  .047 

1  .45  ±  0  .036 

What  is  the  best  value  and  its  probable  error? 

r.4GO  ±  0*.016 


132  METHOD  OF  LEAST  SQUARES. 

89.  In  the  triangulation  of  Lake  Ontario  two  different 
instruments  gave  for  an  angle,  74°  25'  5". 429  ±  0".29   from 
sixteen  readings,  and    74°  25'  4". 611  ±  0''.22    from  twenty- 
four  readings.     Find  the  most  probable  value  of  the  angle 
and  its  probable  error. 

90.  In  each  of  Examples  39-45  find  the  mean  and  prob- 
able errors  and  average  deviation  of  each  observation  and  of 
the  most  probable  value,  using  formulas  from  (50)  to  (63) 
according  as  they  apply. 

91.  In  Example  42  divide  the  observations  in  their  order 
into  six  groups  of  four  observations  each  and  compute  the 
mean  of  each  group.     Then  determine  the  probable  error  of 
the  first  of  these  means  :   ( 1 )  considered  as  a  single  measure 
of  four  times  the  weight  of  those  in  Example  42  ;  (2)  directly 
as  one  of  six  observations  of  equal  weight;    (3)  as  a  deter- 
mination from  its  four  constituents.  0".67  ;  0".72  ;  1".00 

92.  The  following  twenty-nine  measurements  on  the  den- 
sity of  the  earth,  made  by  Cavendish,  give  as  a  mean  result 
5.48.     What  is  the  probable  error  of  an  observation  ?     Solve 
by  the  usual  method  and  also  by  taking   the  residual  that 
occupies  the  middle  position.  0.14 


5.50 

5.55 

5.57 

5.34 

5.42 

5.30 

.61 

.36 

.53 

.79 

.47 

.75 

.88 

.29 

.62 

.10 

.63 

.68 

.07 

.58 

.29 

.27 

.34 

.85 

.26 

.65 

.44 

.39 

.46 

93.  What  is  the  probable  error  of  the  mean  of  two  obser- 
vations which  differ  by  the  amount   a  ? 

94.  A  base-line  is  measxired  five  times  with  a  steel  tape 
reading  to  hundredths  of  a  foot,  and  five  times  with  a  chain 
reading  to  tenths  of  a  foot,  with  results 

By  tape,       741.17       741.09       741.22       741.12       741.10 
By  chain,     741.2         741.4         741.0         741.3         741.1 


EXAMPLES.  133 

Find  the  probable  errors  and  weights  for  a  single  observa- 
tion in  each  case,  and  also  the  adjusted  length  of  the  line 
and  its  probable  error.  741.146  ±  0.015 

95.  Twenty-one  determinations  of  a  chronometer  correc- 
tion gave  results 

-  8.78  -  8.78  -  8.68  -  8.80  -  8.96  -  8.83  -  8.79 
.76  .51  .63  .75  .64  .70  .90 
.85  .64  .58  .78  .65  .64  .93 

Find  the  probable  error  of  the  mean  by  using  both  formulas 
(53)  and  (57),  and  also  determine  the  probable  error  of  a 
single  observation  by  taking  the  middle  residual. 

0.017;    0.018;    0.09 

96.  In  the  following  observations  show  that  M0  =  49.64, 
fi  =  1.95,    r  =  1.31,    /AO  ==  0.40,    r0  =  0.27,    p.3  =  0.87, 
rz  =  0.59. 

M  =    48.81       48.76       49.53       51.56       50.38       49.84 
p    =        5  4  5  3  2  5 

97.  Observations  on  the  time  of  ending  of  a  transit  of 
Mercury  are  made  by  different  observers  with  a  variety  of 
instruments  and  under  more  or  less  favorable  circumstances. 
If  the  weights  assigned  by  the  computer  are  as  indicated,  find 
the  best  value  for  the  time  and  its  probable  error. 

bh  38m  23'     wt.  1  38m  26*     wt.  3  38m  19*     wt.  3 

37  55        "    0  38    21        «    2  38    21        "   2 

38  10       "    1  38    18       «    2           38    15        «   2 

t0  =   5*  38m  19S.9 

98.  •  An  angle  is  measured  five  times  with  a  theodolite,  and 
seven  times  with- a  transit,  giving  results 

Theodolite,    31".7,    39".S,    40".7,    28".6,    32".3 

Transit,          32  .S,    30  .7,    38  .2,    l>9  .3,    41   .6    35".3,    36".2 


134  METHOD  OF  LEAST  SQUARES. 

If  the  relative  values  of  readings  by  the  two  instruments 
are  as  3  to  2,  what  is  the  most  probable  value  of  the  angle  ? 
What  is  the  mean  error  of  the  result  ? 

^99.     Given    J/x    =    65.58   ±  .59,    Jf2    =    35.15   ±  .93, 
M8   =   49.64  ±  .27,    find  the    probable  errors  of   4  J/t  — 

3  J/8  +  2  J/3   and  of   ^  +  ^  -  ^3.  3.69 ;    0.43 

2*  O  T: 

100.  The  three  angles  of  a  triangle  are  measured,  and  the 
probable  error  of  each  observation  is    r  .     What  is  the  prob- 
able error  of  the  triangle  error  ?  r  y/~3~ 

101.  The  zenith  distance  of  a  star  on  the  meridian  is 
observed  to  be     z    =    21°  17'  20'  .3  ±  2".3.     The  declina- 
tion of   the  star  is  given  as     d    =    19°  30'  14".8  ±  0".8. 
What  is  the  latitude  of  the  place  and  its  probable  error  ? 

L    —    z  -f  d   =    40°  47'  35".l  ±  2"4. 

102.  The  zenith  distance   z   of  a  star  at  upper  culmina- 
tion   is   observed  ,n    times,  and   its   zenith    distance    z>    at 
lower  culmination    n'   times.     If   the  latitude    is   given   by 
L   =   90°  --  |  (z  -f-  z'),    and   the    probable   error   of   an 
observation  is    r,    what  is  the  probable  error  of  the  latitude  ? 

103.  The  horizontal  force  necessary  to  start  a  100-pound 
weight  sliding  along  a  table  is  observed  to  be  15.5  ±  0.2 
pounds.     Find  the  probable  error  of  the  coefficient  of  friction. 

104.  If  a  line  is  measured  by  the  continued  application  of 
a  unit  of  measure,  and   r  is  the  probable  error  of  the  placing 
and  reading  of  this  measure,  what  is  the  probable  error  of 
the  length   I  ?  r  f[~ 

105.  If  the  average  deviations  of   z1?  z«,  2s?    are  at  ^»  c> 
respectively,  what  is  the  average  deviation  of  zf  -(-  z22  -\-  zs2? 

106.  If  the  radius  of  a  circle  is  measured   with  result 
1000.0  ±  2.0,    how  should   the   circumference  and  area   be 
expressed  ? 

107.  Two  sides,    a   and    £>,    and  the  included  angle    C  of 
a  triangle   are   measured  with   results    a   =   252.52  ±  .06 


EXAMPLES.  135 

feet,    b  =   300.01  ±   .06   feet,    C   =    42°  13'  00"  ±  30". 
What  is  the  area  and  its  probable  error  ?  25452  ±  9 

108.  Measurements  of  adjacent  sides  of  a  rectangle  gave 
a  ±  r1?    and    b  ±  r2.      What  is  the  probable  error  of  the 
area,  and  for  what   kind  of  a  rectangle  will   this    probable 
error  be  the  least  ? 

109.  If  the  measured  sides  of  a  rectangle  have  the  same 
a.d.,     what  is  the    a.d.    of   the    diagonal   determined    from 
them  ?  Same 

110.  If   the    sides   of   a   rectangle  are  measured  in  the 
manner  indicated  in  Example  104  and  found  to  be   a   and   b, 
wrhat  is  the  probable  error  of  the  area  ? 

111.  The  correction  to  be  applied  to  a  chronometer  is 
found  to  be    -(-  12m  13*.2  ±  08.3.     Ten  days  later  the  cor- 
rection is  again  determined  and  found  to  be    12m  21*.4  ±  0*.3. 
What  is  the  mean  daily  rate  and  its  probable  error  ? 

0*.820  ±  Os.042 

112.  Measurements   of   the    compression    of    the    earth's 
meridian  have  resulted  in 

±  .000046 


294 

What  is  the  probable  error  of  the  denominator  294  ?        3.98 

113.  The    current   flowing    in    a    circuit   is   due    to    two 
sources  whose   electromotive   forces  are  determined    to    be 
«!    =    200  ±2,    e3    =    400  ±  3.     The   resistance  of  the 
circuit  is    30  ±  1.     Find  the  current  and  its  probable  error. 

20  ±  0.68 

114.  The  side    b   and  angles    B   and    C  of  a  triangle  are 
measured    with    results     b    =    106   ±   .06   metres,     H    = 
29°   39'   ±   1',     C    =     120°  7'   ±   2'.      What   is   the    most 
probable  value  of  the  angle    A    and  of  the  side    c  ? 

A    =    30°  14'  ±  2'.2;    c    =    185.5J5  ±  0.15 

115.  The  distance  between  two  divisions  on   a  graduated 
scale  is  measured  by  a  micrometer.     Show  that  the  average 


136  METHOD  OF  LEAST  SQUARES. 

deviation  of  the  mean  of  two  results  is  the  same  as  the  aver- 
age deviation  of  a  single  reading. 

116.  If  the  weights  of  the  determinations  of  three  angles 
A,  B,  C,   are   3,  3,  1,    respectively,  what  is  the  weight  of 
the  sum  of  the  three  angles  ?  0.6 

117.  If  the  weight   of   x   is  />,    what  is  the  weight   of 
loga  *  ? 

118.  If  05    =  —    and  the  weight  of   y   is  p,   what  is  the 

c 
weight  of   x  ?  czp 

119.  In    Example    107,  how   closely  must  the  parts   be 
measured  in  order  to  obtain  the  area  within  0.5  per  cent  ? 

120.  From  observations   on    I  and   t  the  value  of   g   is 
to  be  computed  by  the  pendulum  formula 


t       =      7T   \/  

9 

What  changes  in  g  will  be  produced  by  changes  in  I  and 
t  of  Si  and  82  units,  respectively,  and  what  are  the  allow- 
able errors  in  I  and  t  it  g  is  to  be  determined  within 
1  per  cent  ? 

121.  The  moment  of  inertia  of  a  cylindrical  bar  is  to  be 
obtained  from  measurements  on  its  mass  w,  its  length  A, 
and  its  diameter  d.  The  error  in  the  determination  of  m 
is  negligible,  the  precision  of  the  determination  of  d  is  four 
times  that  of  h.  If  the  measurements  give  m  =  48, 
h  =  8.000,  d  =  1.200  ±  0.10,  and 


T  I  , 

I  =    m  I 

1  12         16 


what  is  the  probable  error  of   I,    and  what  should  be  the 
ratio  of    c?   to    A    to  determine    I  most  accurately  ? 

d  :  h    —    256  :  9 


EXAMPLES.  137 

122.  If   observations  give  for  a  certain  quantity   x  the 
value  303,  with  a  mean  error  of   2,    what  is  the  mean  error 
of  the  expression   3  x  -f-  Iog10  2  x  ? 

123.  The   probable   error    of  the   determination   of   the 
angle   A   is  20".     What  is  the  maximum  probable  error  of 
sin  A  -j-  oos  A  ? 

124.  If  the  probable  error  of  an  observation  on  an  angle 
is    10",    is  there  any  difference  between  the  probable  error 
of  the  function    sin  A  -\-  cos  A  -\-  sin  C  and  of  the  func- 
tion  sin  A  -j-  cos  J5  -f-    sin  (7,    supposing   A    and    B   are 
of  the  same  magnitude  ? 

125.  Given  the  observations, 

Zj  -  2z2  +    z8  -  3    =    0       3*!  -f-    22  -f  228  -  17    =    0 
3z2  -  4z8  -  2    =    0     -  «!  4-  4Z2  -j-  3z,  -  10    =    0 


Find  the  most  probable  values  of   zu  z,,  zs,    and  also  their 
weights  and  precision  measures. 

Z!    ==    3.541  ;   p^    =    29 ;    rZl    =    .024 

126.  Find  the  weights   and    precision    measures   of   the 
unknowns  in  Examples  48  to  57. 

127.  Determine  the  probable  errors  of  the  constants  in 
Examples  72  to  79,  inclusive. 

128.  The  length  of  a  pendulum  which  beats  seconds  is 
given  by 

I    ==    /'  -|-  I  —  q  --  s\  I'  sin'Z 

where   I'   is  the  length  at  the  equator,    q   the  ratio   of 

289 

the  centrifugal  force  at  the  equator  to  the  weight,  and    s   the 
compression  of  the  meridian  regarded  as  unknown.     Putting 


I'    =    991—  +  *,  q  -   s     I'    =    y, 

\  2  I 

observations  in  different  latitude*  gave  in    millimetreR   the 


138  METHOD  OF  LEAST  SQUARES. 

following  equations,  from  which  we  are  to  determine    I  and 
a   together  with  their  probable  errors  :  — 


x  -|-  0.969y  = 

5.13 

x  -f  0.152y  = 

0.77 

x  _|_  0.749y  = 

3.97 

x  +  0.327y  = 

1.70 

x  -f  0.426y  = 

2.24 

x  4-  0.685y  = 

3.62 

x  4-  0.095y  = 

0.56 

*  +  0.793y  = 

4.23 

x         = 

0.19 

I'    =    991.069  ±  .026;       s    =    —  ±  0.00046 

294 

129.  Find  the  weights  and   precision    measures   of   the 
unknowns  in  Examples  64  to  70,  and  also  in  Examples  59  to 
€3,  and  in  71. 

130.  In  Example  95  the  probable  error  of  a  single  obser- 
Tation  is  0.08  seconds.     Find  the  number  of  errors  which 
should  fall  between  0.00  seconds  and  0.10  seconds,  between 
0.10  seconds  and  0.20  seconds,  and  also  the    number   that 
should  be  over  0.20  seconds.     Compare  the  results  with  the 
number  actually  found. 

131.  In   470   determinations  of   the  right  ascensions   of 
Sirius  and  Altair  made  by  Bradley,  the  probable  error  of  a 
single  observation  was  0".2637.    The  number  of  errors  falling 
between  specified  limits  was  as  shown  below.     Compare  this 
result  with  the  distribution  of  errors  called  for  by  the  theory. 

Limits.  Errors.  Limits.  Errors. 

0".0  to  0".l  94  0".6  to  0".7  26 

0  .1  to  0  .2  88  0  .7  to  0  .8  14 

0  .2  to  0  .3  78  0  .8  to  0  .9  10 

0  .3  to  0  .4  58  0  .9.  to  1  .0  7 

0  .4  to  0  .5  51  Over  1  .0  8 

0  .5  to  0  .6  36 

132.  What  is  the  probability  that  the  error  of  a  single 
observation  will  be  as  large  as  twice  the  probable  error? 
As  large  as  five  times  the  probable  error  ? 


EXAMPLES.  139 

133.  On  the  average  how  many  observations  must  be  made 
before  an  error  as  large  as  three  times  the  mean  error  will 
occur  ? 

134.  In  Example  46,  assuming  that  all  errors  between  any 
two  limits  fall  half  way  between  those  limits,   compute  the 
average  deviation  and  mean  error  of  an  observation  and  com- 
pare their  ratio  with  the  theoretical  value  given  in  the  table 
in  paragraph  55. 

135.  A  line  is  measured  500  times  and  the  probable  error 
of  each  observation  is  0.6  cm.     How  many  errors  should 
occur  between  0.4  c.m.  and  0.8  c.m.  ? 

136.  Show  how  the  value    of    -n-    could   be    determined 
experimentally  from  observations  such  as  those  in  Example 
131. 

137.  In  a  system  of  observations  all  equally  good,  r  being 
the  probable  error  of  a  single  observation,  if  two  observations 
are  taken  at  random,  what  quantity  is  their  difference  as 
likely  as  not  to  exceed,  and  what  is  the  probability  that  the 
difference  will  be  less  than   r?  r^lT;    0.367 

138.  In  the  following  measurements  of  an  angle,  ought 
any  of  the  observations  to  be  rejected  ? 

12'  51".75         47".85         47".40         48".90         44".45 
48  .45         51  .05         48  .85         50  .95 
50  .60         47  .75         49  .20         50  .55 

139.  Determine    whether    any    of    the    observations    in 
Example  44  should  be  rejected. 

140.  A  quantity   M  is  measured  with  the  results  given 
below.     Ought  all  the  observations  to  be  retained  ? 

M  =  236,   251,   249,   252,   248,  254,  246,   257,   243,  274 

141.  A   certain  angle  has  been  laid  out  with  such  accu- 
racy that  its  true  value  may  be  taken  as  exactly  90°.    Twenty- 
five  observations  are  made  upon  it  with  a  transit  that  it  is 


140  METHOD  OF  LEAST  SQUARES. 

desired  to  test,  and  the  result  obtained  is  89°  59'  57"  ±  0".8. 
What  are  the  odds  in  favor  of  a  constant  error  in  the  instru- 
ment between  —  1"  and  —  5"?  Between  0"  and  —  6"? 

908  :  92  ;     86  :  1 

142.  Repeated  measurements  of  a  standard  metre  bar 
with  a  decimetre  scale  gave  a  result  10.032  ±  0.010.  What 
are  the  odds  in  favor  of  a  constant  error  in  the  scale  between 

43:7 


143.  Two   determinations   of  the  length  of  a  line  gave 
683.4  ±  0.3   and  684.9  ±  0.3,    respectively.     Show  that  the 
best  value  for  the  length  is    684.15  ±  0.51,    and  that   the 
probable  systematic  error  of  each  determination  is   0.65. 

144.  Two  men    A   and    B   observe  an  angle  repeatedly 
with  the  same  instrument  with  results 

A.  B. 

47°  23'  40"  23'  35"  47°  23'  30"       24'  00" 

23   45  23    40  23  40        23    20 

23    30  23  50 

Is  there  any  relative  personal  error,  and  what  is  the  best 
final  value?  *  47°  23'  38".2  ±  1".6 

145.  Three  independent   determinations  of  the  capacity 
of  a  condenser  made  with   three  different  instruments  gave 
results    42.22    ±    0.21,    43.40    ±    .15    and    44.20    ±    0.18. 
What  is  the  most  probable  value  of  the  capacity  ? 

For  extended  treatment  of  the  subject  illustrated  in  Exam- 
ples 143  to  145  see  Johnson,  "The  Theory  of  Errors  and 
Method  of  Least  Squares,"  chap.  vii. 

146.  In  an  estimation  of  tenths  what  is  the  probable  error 
of  an  observation  ?     What  is  the  average  deviation  ?         0.025 

147.  In  obtaining  the  angle  of  deflection  of  the  needle  of 
a  tangent  galvanometer  by  the  usual  method  what  is  the 
probable  error  of  the  result  ? 


EXAMPLES.  141 

148.  If  all  the  errors  of  a  series  of  observations  must  fall 
between    0    and    a,    and  the  frequency  of  any  error  is  pro- 
portional to   its  magnitude,   what   is  the   Curve   of    Error? 

What  are  the  values  of   r,    a.d.,    and   /*  ?  r    =    - 

V"2~ 

149.  In  Example  148  what  is  the  probability  that  the  error 
of  a  single  observation  will  be  as  large  as   0.5  a. 

150.  If  all  values  of   x  between    0    and   a   are  possible, 
and  their  probabilities  are  proportional  to  their  squares,  find 
the  mean  value  of   x   and  the  probability  that   x   will  be  as 
large  as   0.5  a.     Also  draw  the  Curve  of  Error. 

151.  What  is  the  greatest  probable  error  of  a  logarithm 
found  by  interpolation  in  a  seven  place  table  ?       .000000015 

152.  Given  the  following  set  of  Normal  Equations,  together 
with    [mm]    =    1.3409,    find  the  most  probable  values  of 
the  unknowns  and  their  weights  and  probable  errors.     There 
were  sixteen  observations. 

3.1217  zl  4-     .5756  z2  -     .1565  z3  -  .0067  z< 

-  1.5710      =    0 

.5756  zl  4-  2.9375  z2  -f-     .1103  za  -  .0015  z< 

4-  .9275      =    0 

-  .1565  zl  4-     .1103  z2  4-  4.1273  z3  4  .2051  zt 

+  .0652   =  0 

-  .0067  z,  -  .0015  z2  4-  .2051  z3  -j-  4.1328  z< 

-f  .0178   =  0 
zl  =   0.583  ±  0.018 
z4  =  -  0.004  ±  0.015 


153.     From  ten  observation  equations,  for  which  was  found 
[mm]    =    2.6322,    there  resulted  the  normal  equations 

5.2485  zj   -  -   1.7472  z2  -   2.1954  z,  -f-  0.5399    =    0 

-    1.74722t  -f  1.SS59  za  4-  0.8041  z3   --    1.4493    =    0 

-   2.1954  zt  -j-  0.8041  z2  -j-  4.0440  z8   -    1.S6S1    =    0 


142  METHOD  OF  LEAST  SQUARES. 

Find  the  most  probable  values  of   zx,  z2  and  za   together  with 
their  probable  errors.  z^    •=.    0.42  ±  0.11 

154.  Find  the  most  probable  values  of  the  unknowns  in 
the  normal  equations 

459  zx  —  308  z2  —  389  z3  -f-  244  z4  —  507    =    0 

_  308  zt  -f-  464  z2  -f-  408  zs  —  269  z4  -f  695    =    0 

—  389  «!  4-  408  z2  -f  676  z3  -  381  z4  -f-  653    =    0 

244  zt  —  269  z2  —  331  z3  -f-  469  24  --  283    —    0 

[mm]    =    1129 
z4    =     _  0.488 ;  p,t    =    281 

155.  If  thirteen  observation  equations  give  rise  to   the 
result    [mm~\    =    100.34   and  to  the  normal  equations 

17.50  zt  -     6.50  z2  -     6.50  z3  -     2.14    =    0 

—  6.50  zl  -j-  17.50  z2  —     6.50  z3  —  13.96    =    0 

—  6.50  zt  —     6.50  z2  +  20.50  z3  +     5.40    =    0 

show  that  the  most  probable  values  of  the  unknowns  are 
zt  =  0.67  ±  0.60,     z2  =   1.17  ±  0.60,     28  =   0.32  ±  0.55 


APPENDIX. 


ELEMENTS  OP  THE    THEORY  OP  PROBABILITY. 

200.    Definition.    If  an  event  can  happen  in    a   ways,  and 
fail  in    b   ways,  and  all  these  ways  are  equally  likely  to  occur, 

the  probability  of  the  happening  of  the  event  is  — ^-, 

a    I     o 

and  the  probability  of  its  failure  is   : — -. 

a  -j-  b 

Since  the  event  must  either  happen  or  fail,  the  sum  of  the 
above  probabilities  must  represent  a  certainty.     But 


_    _    -i 

a  -f  b 


That  is,  the  probability  of  a  certainty  is  expressed  by  unity. 
Also,  if  the  probability,  F,  of  the  happening  of  an  event  is 
known,  the  probability  of  its  failure  is  given  at  once  by 
1  -P. 

201.  Example  A.  A  single  throw  is  made  with  a  pair  of 
dice.  What  is  the  probability  that  the  sum  of  the  spots 
turned  up  will  be  5  ? 

Number  of  ways  of  throwing  the  dice  is      6  X  6  ==  36 
Number  of  ways  of  throwing  five  is  4 

4  1 

.-.    Probability  of  throwing  five  is 

36  9 

Example  J3.  A  coin  is  tossed  up  six  times.  Find  the 
chance  that  three  heads  and  three  tails  will  be  the  result 


ii  METHOD   OF  LEAST  SQUAEES. 

Number  of  ways  of  throwing  the  coin  is  2*    =   64 

6x5x4 

Number  of  ways  of  throwing  three  heads  is =   20 

1X2X3 

20  5 

Probability  of  throwing  three  heads  is  —  =  — 

64          16 

202.  Compound  Events.  A  certain  event  can  happen  in 
a  ways  and  fail  in  b  ways  :  a  second  independent  event  can 
happen  in  a'  ways,  and  fail  in  b1  ways,  all  of  these  ways 
being  equally  likely  to  occur. 

To  find  the  probability  of  the  simultaneous  occurrence  of 
the  two  events. 

The  total  number  of  ways  in  which  the  events  can  take 
place  together  is  (a  -\-  b)  (a1  -\-  b') 

(1)  Both  events  can  happen  in    a  a'   ways. 

(2)  Both  events  can  fail  in    b  b'   ways. 

(3)  First  event  can  happen  and  second  fail  in    a  b'    ways. 

(4)  First  event  can  fail  and  second  happen  in    a'  b   ways. 

The  probability  of    (1)    is 


(a  +  b)  (a1  +  b') 

The  probability  of    (2)    is   ; — , ; — ; — rr- 

(a  +  b)  (a1  -j-  b1) 

ft  />' 
The  probability  of    (3)    is 


The  probability  of    (4)    is 


(«  +  *)   («'  +  *') 

aTb 

(a  -f  b)  (a1  +  b') 


But  the  probability  of  the  happening  of  the  first  event  is 
,    and  of  the  second  event  is  —, —  — r/,   etc.     Hence  it 


will  at  once  be  seen  that  the  probability  of  the  simultaneous 
occurrence  of  two  independent  events  is  equal  to  the  product 


APPENDIX.  iii 

of  the  probabilities  of  the  occurrence  of  the  component  events. 

Or,  in  general,  if   Plt  P2,  .  .  .  Pn   are  the  probabilities  of 

the  occurrence  of  any  number,    /?,    of  independent  events,  the 

probability  of  the  simultaneous  occurrence  of  all  the  events  is 

Pl    X    P,    X    ...    Pn  (A) 

By  independent  events  is  meant  those  such  that  the  manner 
of  occurrence  of  one  has  no  influence  upon  the  manner  of 
occurrence  of  the  others. 

203.    Example   C.     The  chance  that   A    can  solve  a  cer- 

2 
tain  problem  is    — ,    and  the  chance  that    B    can  solve  it  is 

Find, 
12 

(a)  The  probability  that  both  will  solve  it. 

(b)  The  probability  that  the  problem  will  be  solved. 

For  (a).  This  is  a  question  as  to  the  probability  of  the 
concurrence  of  two  independent  events.  Therefore  by  an 
application  of  (A),  the  probability  that  both  will  solve  the 
problem  is 

2  N       5  A 

3  '      12  18" 

For  (b).     The  problem  will  be  solved  unless  both  fail. 

The  probability  that  both  will  fail  is  v  —   =    — 

8        12  86 

7  29 

The  probability  of  getting  a  solution  is    1   -         -    =   - 

36  36 

Example  D.  A  pack  of  cards  is  cut,  and  those  taken  off 
then  replaced.  In  how  many  trials  will  it  be  an  even  wager 
that  an  ace  will  be  cut? 


iv  METHOD  OF  LEAST  SQUARES. 

Let    n    be  the  number  of  trials.     Then    n    is  to  be  found 
from 

—\=    - 
52  /  2 

where  the  first  member  of  the  equation  represents  the  proba- 
bility that  we  shall  not  fail   n   times  in  succession. 

Solving  for   n, 

log  2 
n    =    2 

log  52  —   log  48 

=    8.7 

In  nine  trials  then  there  is  a  little  more  than  an  even  chance 
of  cutting  an  ace. 

204.  Dependent  Events.    If  we  have  a  number  of  events 
whose  modes  of  occurrence  are  dependent  one  upon  another, 
the  probability  of  their  concurrence  will  be  found  by  the  same 
method  as  in  paragraph  202  ;    a'   now  denoting  the  number 
of  ways  in  which  after  the  first  event  has  happened   the 
second  will  follow,  and    b1   the  number  of  ways  in  which  after 
the  first  has  happened  the  second  will  not  follow,  etc.    Accord- 
ingly, the  general  formula    (A)    of  paragraph  202  applies  to 
dependent  events  a»  well  as  to  independent  ones. 

Also,  if  an  event  can  take  place  in  a  variety  of  ways,  the 
total  probability  of  its  occurrence  will  be  the  sum  of  the 
probabilities  of  its  occurrence  in  each  of  the  different  ways. 

205.  Example  E.     Suppose  two  purses  contain  respect- 
ively five  dimes  and  a  copper,  and  six  dimes.     A  coin  is  taken 
at  random  from  the  first  purse  and  placed  in  the  second,  and 
then  a  coin  is  transferred  from  the  second  to  the  first.     What 
is  the  probability  that  the  copper  will  remain  in  the  first  purse  ? 

The  probability  that  the  copper  will  be  taken  from  the  first 
purse  and  placed  in  the  second,  and  then  returned  to  the  first 
purse  is 

1_         J_  J_ 

~6    X   Y  42 


APPENDIX. 


and  the  probability  that  the  copper  will  not  be  taken  from 
the  first  purse  at  all  is 

5 

~6~ 

Therefore  the  probability  that  the  copper  will  finally  remain 
in  the  first  purse  is 

_1_          5_  3C  ^ 

AO  R  Af>  1 


FUNCTIONS    OF    SEVERAL    VARIABLES. 

206.  For  the  application  of  Taylor's  Theorem  to  the 
expansion  of  a  function  of  several  independent  variables,  see 
Osborne's  "  Differential  and  Integral  Calculus,"  page  145. 
And  for  the  conditions  that  lead  to  maxima  and  minima 
values  of  such  functions,  see  page  155  of  the  same  work. 


BIBLIOGRAPHY. 


The  following  brief  list  of  treatises,  dealing  with  the 
Method  of  Least  Squares,  is  appended  for  the  benefit  of  those 
whose  professional  work  requires  such  constant  application 
of  the  process  as  to  render  desirable  a  more  detailed  knowl- 
edge of  various  special  methods  of  solution.  In  connection 
with  some  of  the  titles  attention  is  called  to  the  subjects  the 
treatment  of  which  is  particularly  full. 

fohnson,  "  The  Theory  of  Errors  and  Method  of  Least  Squares." 

I'rolxMlity  of  Errvrs.     Systematic  Errors.     The  Method  of  Substitu- 
tion. 

Wright,     "  Treatise  on  the  Adjustment  of  Observations." 

Specinl   Method*  of  Solution.     Applicationt  to  Geodetic  and  Engi- 
neering J'roblemt. 


vi  METHOD  OF  LEAST  SQUARES. 

Merriman,   "  Text-Book  on  the  Method  of  Least  Squares." 

Chauvenet,  "  Treatise  on  the  Method  of  Least  Squares." 

Development  of  the  Theory.    Applications  to  Astronomical  Observa- 
tions. 

Bobek,  "  Lehrbuch  der  Ausglelchsrechnung  nach  dor  Methode 
der  Kleinsten  Quadrate." 

General  Synopsis  of  the  Method,  illustrated  by  Numerous  Examplet. 
Koll,  "  Die  Methode  der  Kleinsten  Quadrate." 

Applications  to  Geodesy. 

Hansen,        "  Von  der  Methode  der  Kleinsten  Quadrate." 
Applications  to  Geodesy. 

Helmert,       "  Die  Ausgleichungsrechnung  nach  der  Methode  der  Klein- 
sten Quadrate." 
Liagre,         "  Calcul  des  Probabilities. " 

Holman,      "  Discussion  of  the  Precision  of  Measurements." 
Problems  in  Physics  and  Electrical  Engineering, 

Weinstein,    "  Handbuch  der  Physikalischen  Maassbestimmungen." 
Applications  to  Physical  Problems. 

Oppolzer,     "  Lehrbuch  znr  Bahnbestimmung  der  Kometen  und  Plane- 

ten." 
Jordan,        "  Handbuch  der  Vermessungskunde." 

For  a  complete  list  of  works  on  the  Method  of  Least 
Squares  published  up  to  1876,  see 

Merriman,   "A  List  of  Writings  relating  to  the  Method   of  Least 
Squares,  with  Historical  and  Critical  Notes." 
Published  in  the  Transactions  of  the  Connecticut  Academy,  vol.  iv, 
1877. 

Notice  of  works  published  since  1876  may  be  found  in 
periodicals  devoted  to  the  progress  of  Mathematical  Science. 
Such  as 

"  Jahrbuch  (iber  die  Fortschritte  der  Mathematik." 
"  Bulletin  des  Sciences  Mathematiques." 


TABLES. 


Vll 


TABLE  I. 
Values  of  the  Integral  —  I    e-^dt  for  Argument 


t  a 

or   — 

0.4769  r 


a 
r 

01234 

56789 

Diff. 

0.0 
0.1 
0.2 
0.3 
0.4 

0.0000  0.0054  0.0108  0.0161   0.0215 
0538      0591      0645      0699      0752 
1073      1126      1180      1233      1286 
1603      1656      1709      1761      1814 
2127      2179      2230      2282      2334 

0.0269  0.0323  0.0377  0.0430  0.0484 
0806      0859      0913      0966      1020 
1339      1392      1445      1498      1551 
18C6      1!H8      1971      2023      2075 
2385      2436      2488      2539      2590 

54 
54 
53 
52 

51 

0.5 
0.6 
0.7 
0.8 
0.9 

0.2641   0.2691   0.2742  0.2793  0.2843 
3143      3192      3242      3291      3340 
3632      3680      3728      3775      3823 
4105      4152      4198      4244      4290 
4562      4606      4651      4695      4739 

0.2893  0.2944  0.2994  0.3043  0.3093 
3389      3438      3487      3535      3583 
3870      3918      3965      4012      4059 
4336      4381      4427      4472      4517 
4783      4827      4860      4914      4957 

50 
49 
46 
45 
43 

1.0 
1.1 
1  2 
1.3 
1.4 

0.5000  0.5043  0.5085  0.5128  0.5170 
5419      5460      5500      5540      5581 
5817      5856      5894      5!)32      5970 
6194      6231      6267      6303      6339 
6550      6584      6618      6652      6686 

0.5212  0.5254  0.5295  0.5337  0.5378 
5620      5660      5700      5739      5778 
0008      6046      6083      6120      6157 
6375      6410      6445      6480      6515 
6719      6753      6786      6818      6851 

41 

3!» 

37 
35 
32 

1.5 
1.6 
1.7 

1  8 

1.9 

0.6883  0.6915  0.6947  0.6979  0.7011 
7195      7225      7255      7284      7313 
7-185      7512      7540      7567      7594 
7753      7778      7804      7829      7854 
8000      8023      8047      8070      8093 

0.7042  0.7073  0.7104  0.7134  0.7165 
7342      7371      7400      7428      7457 
7621      7648      7675      7701      7727 
7879      7904      7928      7952      7976 
8116      8138      8161      8183      8205 

30 
28 
26 
24 
22 

20 
•i  \ 
22 
23 
2.4 

0.8227  0.8248  0.8270  0  8291   0.8312 
8433      8453      8473      84i»2      8511 
8622      8039      8657      8674      8«»2 
8792      8808      8824      8840      8855 
8945      8960      8974      8988      9002 

0.8332  0.8353  0.8373  0.8394  0.8414 
8530      8549      8567      8585      8604 
8709      8726      8742      8759      8775 
8870      8886      8901      8916      8930 
9016      9029      9043      9056      9069 

19 
18 
17 
15 
13 

2.5 
2.6 
2.7 
2.8 
2.9 

0.90S2  0.9095  0.9108  0.9121   0.9133 
9205      9217      9228      9239      9250 
9314      9324      9334      9344      9ar>4 
9410      9419      9428      9437      9446 
9495      9503      9511      9519      9526 

O.f'146  0.9158  0.9170  0.9182  0.9103 
9261      9272      9283      9293      9304 
9.'!64      9373      9383      9392      9401 
9454      9403      9471      9479      94H7 
9534      9541      9548      9556      9563 

12 
10 
9 
8 

7 

30 
3.1 
3.2 
33 
3.4 

0.9570  0.9577  0.9583  0.9590  0.95!>7 
9635      9U41       9647      9C52      9658 
9691      9606      9701      9706      9711 
9740      9744      974'.)      9753      9757 
9782      9786      9789      9793      9797 

0.9603   0.9610  O.P616   0.9T.22  O.W29 
IM'4J4      9669      9675      9ti80      9H86 
9716      !>72l      9726      9731      9735 
97fil       97C6      9770      97:4      9778 
9800      9804      9807      9811      9814 

6 
5 
5 
4 
4 

3. 

0.9570  0.9635  0.9691   0.9740  0.9782 

0.9818  0.9848  0.9874   0.9896  0.9915 

4. 

9930      9943      9954      9963      9970 

997C      9981      9US5      9988      9900 

5. 

9993      9994      9996      9997      9997 

9998      9998      9999      9999      9999 

CO 

1.0000 

01234 

r 

50789 

OUT. 

Vlll 


METHOD  OF  LEAST  SQUARES. 


TABLE  II. —  Common  Logarithms. 


n 

0             1234 

56789 

Diflf. 

10 
11 
12 
13 
14 

0000      0043      0086      0128      0170 
0414      0453      0402      0531      0569 
0792      0828      0864      0899      0034 
1139      1173      1206      1239      1271 
1461      1492      1523      1553      1584 

0212      0253      0294      0334      0374 
0607      0645      0682      0719      0756 
0<J69      1004      1038      1072      1106 
1303      1335      1367      1399      1430 
1614      1644      1673      1703      1732 

42 
38 
35 
32 
30 

15 
16 
17 
18 
19 

1761      1790      1818      1847      1875 
2041      2068      2095      2122      2148 
2304      2330      2356      2380      2406 
2553      2577      2601      2625      2648 
2788      2810      2833      2856      2878 

1903      1931      1959      1987      2014 
2175      2201      2227      2263      2279 
2430      2455      2480      2604      2529 
2672      2695      2718      2742      2765 
2900      2923      2945      2967      2989 

28 
27 
25 
24 
22 

20 
21 
22 
23 

24 

3010      3032      3054      3075      3096 
3222      3243      3263      3284      3304 
3424      3444      3464      3483      3502 
3617      3636      3C55      3674      3C92 
3602      3820      3838      3856      3874 

3118      3139      3160      3181      3201 
3324      3345      3365      3385      3404 
3522      3541      3560      3579      3598 
3711      3729      3747      3766      3784 
3892      3909      3927      3945      3962 

21 
20 
19 
18 
18 

25 
26 
27 
28 
29 

3979      3997      4014      4031      4048 
4150      4166      4183      4200      4216 
4314      4330      4346      4362      4378 
4472      4487      4502      4518      4533 
4624      4639      4654      4669      4683 

4065      4082      4099      4116      4133 
4232      4249      4265      4281      4298 
4393      4409      4426      4440      4456 
4548      4564      4579      4594      4(i09 
4698      4713      4728      4742      4757 

17 
17 
16 
15 
15 

30 
31 
32 
33 
34 

4771      4786      4800      4814      4829 
4914      4928      4942      4955      49t9 
5051      5065      5079      5092      6105 
5185      5198      6211      5224      5237 
5315      5328      6340      5353      5366 

4843      4857      4871      4886      4900 
4983      4997      6011      6024      5038 
6119      6132      5145      5159      6172 
5250      5263      5276      5289      5302 
6378      5391      5403      5416      6428 

14 
14 
13 
13 
13 

35 
36 
37 
38 
39 

6441      5453      6465      5478      5490 
6563      5575      5587      5599      5611 
6682      5694      6705      5717      6729 
6798      5809      6821      6832      6843 
6911      5922      6933      5944      6955 

5502      6514      5527      5539      5551 
6623      6635      5647      5658      5670 
6740      6752      6763      5775      6786 
5855      6866      6877      6888      5809 
59C6      5977      5988      5999      6010 

12 
12 
12 
11 
11 

40 
41 
42 
43 
44 

6021      6031      6042      6053      6064 
6128      6138      6149      6160      6170 
6232      6243      6253      6263      6274 
6335      6345      6355      6365      6375 
6435      6444      6454      6464      6474 

6075      6085      6096      6107      6117 
6180      6191      6201      6212      6222 
6284      6294      6304      6314      6325 
6385      6395      6405      6415      6425 
6484      6493      6503      6513      6522 

11 
11 
10 
10 
10 

45 
46 
47 
48 
49 

6532      6542      6551      6661      6571 
6628      6637      6646      6666      6665 
6721      6730      6739      6749      6758 
6812      6821      6830      6839      6848 
6902      6911      6920      6928      6937 

6580      6590      6599      6609      6618 
6675      6684      6693      6702      6712 
6767      6776      6785      6794      6803 
6857      6866      6875      68«4      6893 
6946      6956      6964      6972      6981 

10 
9 
9 
9 
9 

50 
51 
52 
53 
54 

6990      6998      7007      7016      7024 
7076      7084      7093      7101      7110 
7160      7168      7177      7186      7193 
7243      7251      7259      7267      7275 
7324      7332      7340      7348      7356 

7033      7042      7050      7059      7067 
7118      7126      7135      7143      7152 
7202      7210      7218      7226      7235 
7284      7292      7300      7308      7316 
7364      7372      7380      7388      7396 

9 
8 
8 
8 
8 

n 

01234 

56789 

Diff. 

TABLES. 


TABLE  II. —  Common  Logarithms. 


n 

01234 

56789 

Diff. 

55 
56 
67 
58 
59 

7404   7412   7419   7427   7435 
7482   7490   7497   7505   7513 
7559   7566   7574   7582   7589 
7634   7642   7649   7657   7664 
7709   7716   7723   7731   7738 

7443   7451   7459   7466   7474 
7520   7528   7536   7543   7551 
7597   7604   7612   7619   7627 
7672   7679   7686   7694   7701 
7745   7752   7760   7767   7774 

8 

60 
61 
62 
63 
64 

7782   7789   7796   7803   7810 
7853   7860   7868   7875   7882 
7924   7931   7938   7945   7952 
7993   8000   8007   8014   8021 
8062   8069   8075   8082   8089 

7818   7825   7832   7839   7846 
7889   7896   7903   7910   7917 
7959   7966   7973   7980   7987 
8028   8035   8041   8048   8055 
8096   8102   8109   8116   8122 

7 

65 
66 
67 
68 
69 

8129   8136   8142   8149   8156 
8195   8202   8209   8215   8222 
8261   8267   8274   8280   8287 
8325   8331   8338   8344   8351 
8388   8395   8401   8407   8414 

8162   8169   8176   8182   8189 
8228   8235   8241   8248   8254 
8293   8299   8306   8312   8319 
8357   8363   8370   8376   8382 
8420   8426   8432   8439   8445 

7 

70 
71 
72 
73 
74 

8451   8457   8463   8470   8476 
8513   8519   8525   8531   8537 
8573   8579   8585   8591   8597 
8633   8639   8645   8651   8657 
8692   8698   8704   8710   8716 

8482   8488   8494   8500   8606 
8543   8549   8555   8561   8567 
8603   8609   8615   8621   8627 
8663   8669   8675   8681   8686 
8722   8727   8733   8739   8745 

6 

75 
76 

77 
78 
79 

8751   8756   8762   8768   8774 
8808   8814   8820   8825   8831 
8S65   8871   8876   8882   8887 
8921   8>J27   8932   8938   8943 
8976   8982   8987   8993   8998 

8779   8785   8791   8797   8802 
8837   8842   8848   8854   8859 
8893   8899   8904   8910   8915 
8949   8954   8960   8965   8971 
9004   9009   9015   9020   9025 

6 

80 

81 
82 
83 
84 

9031   9036   9042   9047   9053 
9085   9090   9096   9101   9106 
9138   9143   9149   9154   9159 
9191   9196   9201   9206   9212 
9243   9248   9253   9258   9203 

9058   9063   9069   9074   9079 
1)112   9117   9122   9128   9133 
9166   9170   9175   9180   9186 
9217   9222   9227   9232   9238 
9269   9274   9279   9284   9289 

5 

85 

86 
87 
88 
89 

9294   9299   9304   9309   9315 
9345   9350   9355   9360   9365 
9395   9400   9405   9410   9415 
9445   9450   9455   9460   9465 
9494   9499   9504   9509   9613 

9320   9325   9330   9335   9340 
9370   9375   9380   9385   9390 
9420   9425   9430   9435   9440 
9469   9474   9479   9484   9489 
9518   9523   9528   9533   9538 

5 

90 
91 
92 
93 
94 

9542   9547   9552   9557   9562 
9590   9595   9600   9605   %09 
9638   9643   9647   9652   9657 
9685   9089   9694   9699   9703 
9731   9736   9741   9745   9750 

9566   9571   9576   9581   9586 
9614   9619   9624   9628   9633 
9661   9666   9671   9675   9680 
•J708   9713   9717   9722   9727 
9754   9759   9763   9768   9773 

5 

95 
96 
»7 
M 
99 

9777   9782   9786   9791   9795 
9823   9827   9832   9836   9841 
9868   9872   9877   9881   9886 
9912   9917   9921   9926   9930 
9956   9961   9965   9969   9974 

9800   9805   9809   9814.  9818 
9845   9850   9854   98-r>9   9863 
9890   9894   9899   9903   9008 
9934   9939   9943   9948   9952 
9978   9983   9987   9991   9996 

4 

n     0      1234 

56789 

Diff. 

METHOD  OF  LEAST  SQUAEES. 


TABLE  III.  — Squares  of  Numbers. 


n 

01234 

56789 

Diff. 

1.0 

1.1 

1.2 
1.3 
1.4 

t.OOO     1.020     1.040     1.061      1.082 
1.210     1.232     1.254     1.277     1.300 
1.440     1.464     1.488     1.513     1.538 
1.6HO     1.716     1.742     1.769     1.796 
1.960     1.988     2.016     2.045     2.074 

1.103     1.124     1.145     1.166      1.188 
1.323     1.346     1.369     1.392     1.410 
1.663      1.588     1.613     1.638     1.664 
1.823     1.850     1.877     1.904     1.932 
2.103     2.132     2.161      2.190     2.220 

22 
24 
26 
28 
30 

1.5 
1.6 
1.7 
1.8 
1.9 

2.250     2.280     2.310     2.341      2.372 
2.560     2.592     2.624     2.657     2.690 
2.890     2.924     2.958     2.993     3.028 
3.240     3.276     3.312     3.349     3.38« 
3.610     3.648     3.686     3.725     3.764 

2.403     2.434     2.465     2.496     2.528 
2.723     2.756     2.789     2.822     2.856 
3.063     3.098     3.133     3.168     3.204 
3.423     3.460     3.497     3.534     3.572 
3.803     3.842     3.881     3.920     3.960 

32 
34 
36 
38 
40 

2.0 
2.t 
2.2 
2.3 
2.4 

4.000     4.040     4.080     4.121      4.162 
4.410     4.452     4.494     4.537     4.580 
4.840     4.884     4.928     4.973     5.018 
5.290     5.336     6.382     5.429     5.476 
5.760     5.808     5.856     5.905     5.954 

4.203     4.244     4.285     4.326     4.368 
4.623     4.666     4.709     4.752     4.796 
5.063     5.108     5.153     5.198     5.244 
5.523     5.570     5.617     5.664     6.712 
0.003     6.052     6.101      6.150     6.200 

42 
44 
46 
48 
50 

2.5 
2.0 
2.7 
2.8 
2.9 

6.250     6.300     6.350     6.401      6.452 
6.760     6.812     6.864     6.917     6.970 
7.290     7.344      7.398     7.453     7.508 
7.840     7.896     7.952     8.009     8.066 
8.410     8.468     8.526     8.585     8.644 

6.503     6.554     6.605     6.656     6.708 
7.0^3     7.076     7.129     7.182     7.236 
7.563     7.618     7.673     7.728     7.784 
8.123     8.180     8.237     8.294     8.352 
8.703     8.762     8.821      8.880     8.940 

52 
64 
66 

68 
60 

3.0 
3.1 
32 
3.3 
3.4 

9.000     9.060     9.120     9.181      9.242 
9.610     9.672     9734     9.797     9.860 
10.24     10.30     10.37     10.43     10.50 
10.89     1096     11.02     11.09     11.16 
11.56     11.63     11.70     11.76     11.83 

9.303     9.364     9.425     9.486     9.548 
9.923     9.986     10.05     10.11      10.18 
10.56      10.63      10.69     10.76      10.82 
11.22     11.29     11.36     11.42     11.49 
11.90     11.97     12.04     12.11      12.18 

62 
6 

7 
7 

7 

3.5 
3.6 
3.7 
3.8 
3.9 

12.25     12.32     12.39     12.46     12.53 
12.96     13.03     13.10     13.18     13.25 
13.69     13.76     13.84     13.91      13.99 
14.44     14.52     14.59     14.67     14.75 
16.21      15.29     15.37      15.44     15.62 

12.60     12.67      12.74     12.82     12.89 
13.32     13.40     13.47     13.54      13.62 
14.06     14.14     14.21      14.29     14.36 
14.82     14.90     14.98     15.05     15.13 
15.60     15.68     15.76      15.84     15.92 

7 
7 
8 
8 
8 

4.0 
4.1 
4.2 
43 
4.4 

16.00     16.08      16.16     16.24     16.32 
16.81      16.89     16.97      17.06     17.14 
17.64     17.72     17.81      17.89     17.98 
18.49     18.58      18.66      18.75     18.84 
19.36      19.45     19.54     19.62     l'J.71 

16.40     16.48     16.56     16.65     16.73 
17.22     17.31      17.39     17.47     17.56 
18.06     18.15     18.23     18.32     18.40 
18.92     19.01      19.10     19.18     19.27 
19.80     19.89     19.98     20.07     20.16 

8 
8 
9 
9 
9 

4.5 
4.6 
4.7 
4.8 
4.9 

20.25     20.34     20.43     20.52     20.61 
21.16     21.25     21.34     21.44     21.53 
22.09     22.18     22.28     22.37     22.47 
23.04     23.14     23.23     23.33     23.43 
24.01     24.11     24.21      24.30     24.40 

20.70     20.79     20.88     20.98     21.07 
21.62     21.72     21.81      21.90     22.00 
22.56     22.66     22.75     22.85     22.94 
23.52     23.62     23.72     23.81      23.91 
24.60     24.60     24.70     24.80     24.90 

9 
9 
10 
10 
10 

5.0 
5.1 
5.2 
5.3 
5.4 

25.00     25.10     25.20     25.30     25.40 
26.01      26.11      26.21      26.32     26.42 
27.04     27.14     27.25     27.35     27.46 
28.09     28.20     28.30     2841      28.52 
29.16     29.27     29.38     29.48     29.59 

25.50     25.60     25.70     25.81      25.91 
26.62     26.63     26.73     26.83     26.94 
27.56     27.67     27.77     27.88     27.98 
28.62     28.73     28.84     28.94     29.05 
29.70     29.81      29.92     30.03     30.14 

10 
10 

n 
n 
n 

H 

01              234 

56789 

Diff. 

TABLES. 


TABLE  III.— Squares  of  Numbers. 


n 

01              234 

56             789 

Diff. 

5.5 
5.6 
5.7 
58 
5.9 

30.25     30.36     30.47     30.58     3069 
31.36     31.47     31.58     31.70     31.81 
32.49     32.60     32.72     32.83     32.95 
33.64     33.76     33.87     33.99     34.11 
34.81     34.93     35.05     35.16     35.28 

30.80     30.91      31.02     31.14     31.25 
31.92     32.04     32.15     32.26     32.38 
33.06     33.18     33.29     33.41      33.52 
34.22     34.34     34.46     34.57     34.69 
35.40     35.52     35.64     35.76     35.88 

11 
11 
12 
12 

12 

6.0 
6.1 
6.2 
6.3 
6.4 

36.00     36.12     36.24     36.36     36.48 
37.21     37.33     37.45     37.58     37.70 
38.44     38.56     38.69     38.81     38.94 
39.69     39.82     39.94     40.07     40.20 
40.96     41.09     41.22     41.34     41.47 

36.60     36.72     36.84     36.97     37.09 
37.82     37.95     38.07     38.19     38.32 
39.06     39.19     39.31      39.44     39.56 
40.32     40.45     40.58     40.70     40.83 
41.60      41.73     41.86     41.99     42.12 

12 
12 
13 
13 
13 

6.5 
6.6 
6.7 
6.8 
6.9 

42.25     42.38     42.51      42.64     42.77 
43.56     43.69     43.82     43.96     44.09 
44.89     45.02     45.16     45.29     45.43 
46.24     46.38     46.51      46.65     46.79 
47.61      47.75     47.89     48.02     48.16 

42.90     43.03     43.16     43.30     43.43 
44.22     44.36     44  49     44.62     44.76 
45.56     45.70     45.83     45.97     46.10 
46.92     47.06     47.20     47.33     47.47 
48.30     48.44     48.58     48.72     48.86 

13 
13 
14 
14 
14 

7.0 
7.1 
7.2 

49.00     49.14     49.28     49.42     49.56 
50.41      50.55     50.69     50.84     50.98 
51.84     61.  U8     52.13     52.27     5242 

49.70     49.84     49.98     50.13     50.27 
51.12     51.27     51.41      51.55     51.70 
52.56     52.71     52.85     53  00     53.14 

14 
14 

15 

7.3 

7.4 

53.29     53.44     53.58     53.73     53.88 
64.76     64/J1     55.06     55.20     55.35 

54.02     54.17     64.32     54.46     54.61 
55.50     65.65     65.80     55.95     66.10 

15 
15 

7.5 
7.6 

7.7 
7.8 
7.9 

56.25     56.40     56.55     56.70     56.85 
57.76     57.91      58.06     58.22     58.37 
59.29     59.44     59.60     59.75     59.91 
60.84     61.00     61.15     61.31      61.47 
62.41      62.57     62.73     62.88     63.04 

57.00     57.15     57.30     57.46     57.61 
58.52     58.68     58.83     58.98     59.14 
60.06     60.22     60.37     60.53     60.68 
61.62     61.78     61.94     6_'.09     62.25 
63.20     63,36     63.52     63.68     63.84 

15 
15 
16 
16 
16 

8.0 
8.1 
8.2 
8.3 
8.4 

64.00     64.16     64.32     64.48     64.64 
65.61      65.77      65.93      66.10      66.26 
07.24     67.40     67.57      67.73     67.90 
68.89     69.06     69.22     69.39     69.56 
70.56     70.73     70.80     71.06     71.23 

64.80     64.96     65.12     65.29     65.45 
66.42     66.59     66  75     66.91      67.08 
68.06     68.23     68.39     68.56     68.72 
69.72     69.89     70.06     70.22     70.39 
71.40     71.57      71.74     71.91     72.08 

16 
16 
17 
17 
17 

8.5 
8.6 

8.7 
88 
8.9 

72.25     72.42     72.59     72.76     72.93 
73.96     74.13     74.30     74.48     74.65 
75.69     75.86     76.04     76.21      76.39 
77.44     77.62     77.79     77.97     78.15 
79.21      79.39     79.57     79.74     79.92 

73.10      73.27     73.44     73.62     73.79 
74.82     75.00     75.17     75.34     75.52 
76.56     76.74     76.91      77.09     77.26 
78.32     78.50     78.68     78.85     79.03 
80.10     80.28     80.46     80.64     80.82 

17 
17 

18 
18 
18 

9.0 
9.1 
9.2 
9.3 
94 

81.00     81.18     81.36     81.54     81.72 
82.81      82.99     83.17     83.36     83.54 
84.64     8482     85.01      85.19     85.38 
86.49     86.68     86.86     87.05     87.24 
88.36     88.55     88.74     88.92     89.11 

81.90     82.08     82.26     82.45     82.63 
83.72     83.91      84.09     84.27     84.40 
85.56      85.75      85.93      86.12      86.30 
87.42     87.61      87.80     87.98     88.17 
89.30     89.49     89.68     89.87     90.06 

18 
18 
111 
I'.t 
19 

9.5 
9.6 
9.7 
9.8 
9.9 

90.25     00.44     90.63     90.82     91.01 
92.16     92.35      92.54      92.74      92.93 
94.09      94.28      94.48      94.67      94.87 
96.04      96.24      96.43      96.63      96.RT 
98.01      98.21      98.41      98.60     98.80 

91.20     91.39     91.58     91.78     91.97 
93.12      93.32      93.51      93.70      93.90 
95.06      95.26      95.45      95.65      95.84 
97.02     97.22     97.42     97.61      97.81 
99.00      99.20      9U.40      99.60      99.80 

19 
lit 
20 
20 
20 

n 

01              234 

56789 

DilT. 

1388 


UCLA-Geology/Geophysics  Library 

QA  275  B28g  1915 


L  006  556  080  7 


A""  "•"  Ml"  I    Ml  Hill  III  I  HI 
001  001  075     9 


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